In most practical image segmentation tasks, the image to be segmented will need to first be reconstructed from indirect, damaged, and/or noisy observations. Traditionally, this reconstruction-segmentation task would be done in sequence: first apply the reconstruction method, and then the segmentation method. Joint reconstruction-segmentation is a method for using segmentation and reconstruction techniques simultaneously, to use information from the segmentation to guide the reconstruction, and vice versa. In this talk, we will demonstrate how this can be done using the graph-PDE-based segmentation techniques developed by (Bertozzi, Flenner, 2012) and (Merkurjev, Kostic, Bertozzi, 2013), with ideas drawn from (Budd, Van Gennip, 2020) and (Budd, Van Gennip, Latz, 2020).