In many biomedical imaging applications, we want to construct a high quality 3-dimensional image from a collection of 2-D projection images. The relationship between the multiple 2-D image projections and the 3-D object can be modeled as a separable, nonlinear least squares problem. Difficulties arise because the underlying mathematical model is an ill-posed inverse problem, that is, small inaccuracies and artifacts introduced during the image acquisition process may result in severe errors in the reconstruction. Thus, we develop a Gauss-Newton algorithm, with Tikhonov regularization, for joint estimation of the 3-D image and the parameter errors.
Nonlinear least squares problems of this form are becoming increasingly prevalent in many medical imaging applications, and we will look at one such example from breast cancer screening. Digital tomosynthesis is a promising alternative to Computed Tomography (CT) for 3-D breast imaging, but image reconstruction and post-processing algorithms will be essential to making it clinically viable. The mathematical model for breast tomosynthesis is a nonlinear least squares problem, so we will illustrate the success of our joint optimization approach on examples from this application.