An Implemention of Two-Phase Image Segmentation using the Split Bregman Method

In this paper, we describe an implementation of the two-phase image segmentation algorithm proposed by Goldstein, Bresson, Osher in \cite{gold:bre}. This algorithm partitions the domain of a given 2d image into foreground and background regions, and each pixel of the image is assigned membership to one of these two regions. The underlying assumption for the segmentation model is that the pixel values of the input image can be summarized by two distinct average values, and that the region boundaries are smooth. Accordingly, the model is defined as an energy in which the variable is a region membership function to assign pixels to either region, originally proposed by Chan and Vese in \cite{chan:vese}. This energy is the sum of image data terms in the regions and a length penalty for region boundaries. Goldstein, Bresson, Osher modify the energy of Chan-Vese in \cite{gold:bre} so that their new energy can be minimized efficiently using the split Bregman method to produce an equivalent two-phase segmentation. We provide a detailed implementation of this method \cite{gold:bre}, and document its performance with several images over a range of algorithm parameters.

Probabilistic U-Net with Kendall Shape Spaces for Geometry-Aware Segmentations of Images

One of the fundamental problems in computer vision is image segmentation, the task of detecting distinct regions or objects in given images. Deep Neural Networks (DNN) have been shown to be very effective in segmenting challenging images, producing convincing segmentations. There is further need for probabilistic DNNs that can reflect the uncertainties from the input images and the models into the computed segmentations, in other words, new DNNs that can generate multiple plausible segmentations and their distributions depending on the input or the model uncertainties. While there are existing probabilistic segmentation models, many of them do not take into account the geometry or shape underlying the segmented regions. In this paper, we propose a probabilistic image segmentation model that can incorporate the geometry of a segmentation. Our proposed model builds on the Probabilistic U-Net of \cite{kohl2018probabilistic} to generate probabilistic segmentations, i.e.\! multiple likely segmentations for an input image. Our model also adopts the Kendall Shape Variational Auto-Encoder of \cite{vadgama2023kendall} to encode a Kendall shape space in the latent variable layers of the prior and posterior networks of the Probabilistic U-Net. Incorporating the shape space in this manner leads to a more robust segmentation with spatially coherent regions, respecting the underlying geometry in the input images.