In this work we develop a framework to detect structural variants (SVs) in the genomes of related individuals. In particular, we consider low-coverage regimes that are more inexpensive than in high-coverage settings but are more susceptible to sequencing errors. To improve our ability to accurately predict SVs, we incorporate statistical models with familial relationship constraints and sparsity promoting penalties. We use simulated data to run experiments. Previous detection methods have used Poisson statistical models. The main contribution of this thesis is the use of the more general negative binomial distribution model in one-parent/one-child and two-parent/one-child frameworks. We extend the existing SPIRAL algorithm, which uses a Poisson log-likelihood objective function, and implement a negative binomial log-likelihood objective function. The genomes tested are haploid, meaning there is only one copy of each chromosome.