Microwave inverse scattering algorithms form maps of the dielectric properties of objects. The dielectric properties of objects consist of the permittivity and conductivity. Images are formed through an optimization procedure that compares the electric fields scattered off an object in experiment to the electric fields predicted by a numeric forward model and object. If the numeric forward model and object predicted fields are similar enough to the measured fields, then the numeric object is assumed to be close to the actual object. The forward model contains a complete description of the physics of microwave propagation. There is evidence that benign and malignant breast lesions have different dielectric properties, so if imaged, could provide another method for diagnosis complementary to X-ray mammography or ultrasound. Dielectric properties can only be quantitatively imaged using inverse scattering algorithms. Inverse scattering algorithms are computationally intensive. They require a forward model, a wave simulator (depending on the algorithm), an optimization procedure to minimize a cost-function measuring the difference between measured and predicted fields, and, in experiment, a model of the antennas used to transmit and receive the probing electric fields. In a setup, multiple antennas are arranged around the breast, which is positioned downward in a liquid coupling medium. Different combinations of antennas are used to transmit probing electric fields and receive scattered fields. This algorithm is based on Born iterations. First, the object field is assumed to be the incident field (Born approximation). The incident field is given from the model of the antennas. The field is assumed constant in the forward model. Second, a multi-objective covariance-based cost function is minimized in order to estimate the object with the given field. The cost function is minimized through a conjugate gradient routine; no matrix system is ever formed or solved. Third, the object estimate is used in a wave simulator (forward solver) to obtain a new guess of the object field. The wave simulator can be any full-wave electromagnetic solver; this algorithm uses the Neumann series solution, which enables the solution of large-domain problems. Last, these steps are iterated until the change in the object between iterations is small. The cost function has the probabilistic interpretation of assigning Gaussian prior distributions around the measured data and image pixels. The standard deviations of the data Gaussians are the measured experimental noise. The standard deviations of the image pixel Gaussians are determined by our prior knowledge of the known range of values. These distributions act to regularize the inverse problem. Included in the algorithm is a full-wave description of the antennas acting in both transmitting and receiving modes which allows us to directly link the quantities in the algorithm (fields) to the measurements (S-parameters). This algorithm is important because it includes all the physics of wave propagation to produce quantitative images of breast tissue. The algorithm is stable, robust, has physically meaningful regularization (a feature lacking in many algorithms), avoids storing and solving a large matrix systems, and has been specifically designed for realistically large imaging domains. This imaging algorithm was designed with the features to make quanitative microwave breast imaging possible, without which quanitiative imaging probably cannot succeed in practice.