text: We present a novel method to explicitly incorporate topological prior knowledge into deep learning based segmentation, which is, to our knowledge, the first work to do so. Our method uses the concept of persistent homology, a tool from topological data analysis, to capture high-level topological characteristics of segmentation results in a way which is differentiable with respect to the pixelwise probability of being assigned to a given class. The topological prior knowledge consists of the sequence of desired Betti numbers of the segmentation. As a proof-of-concept we demonstrate our approach by applying it to the problem of left-ventricle segmentation of cardiac MR images of 500 subjects from the UK Biobank dataset, where we show that it improves segmentation performance in terms of topological correctness without sacrificing pixelwise accuracy.
  @cite_19: We introduce a novel algorithm for segmenting the high resolution CT images of the left ventricle (LV), particularly the papillary muscles and the trabeculae. High quality segmentations of these structures are necessary in order to better understand the anatomical function and geometrical properties of LV. These fine structures, however, are extremely challenging to capture due to their delicate and complex nature in both geometry and topology. Our algorithm computes the potential missing topological structures of a given initial segmentation. Using techniques from computational topology, e.g. persistent homology, our algorithm find topological handles which are likely to be the true signal. To further increase accuracy, these proposals are measured by the saliency and confidence from a trained classifier. Handles with high scores are restored in the final segmentation, leading to high quality segmentation results of the complex structures.
  @cite_14: Topological tools provide features about spaces, which are insensitive to continuous deformations. Applied to images, the topological analysis reveals important characteristics: how many connected components are present, which ones have holes and how many, how are they related one to another, how to measure them and find their locations. We show in this paper that the extraction of such features by computing persistent homology is suitable for grayscale image segmentation.
  @cite_12: Automated tumor segmentation in Hematoxylin & Eosin stained histology images is an essential step towards a computer-aided diagnosis system. In this work we propose a novel tumor segmentation approach for a histology whole-slide image (WSI) by exploring the degree of connectivity among nuclei using the novel idea of persistent homology profiles. Our approach is based on 3 steps: 1) selection of exemplar patches from the training dataset using convolutional neural networks (CNNs); 2) construction of persistent homology profiles based on topological features; 3) classification using variant of k-nearest neighbors (k-NN). Extensive experimental results favor our algorithm over a conventional CNN.
summary: Persistent homology has previously been applied to the problem of semantic segmentation, such as in @cite_19 @cite_12 @cite_14 . The important distinction between our method and these previous works is that they apply PH to the input image to extract features, which are then used as inputs to some other algorithm for training. Such approaches can capture complex features of the input images but require those topological features to be directly extractable from the raw image data. Our approach instead processes the image with a CNN and it is the output of the CNN, representing the pixelwise likelihood of the structure we want to segment, which has PH applied to it.