In this paper, we develop a new approach to tomographic reconstruction problems based on geometric curve evolution techniques. We use a low order parametric model to describe the shape and texture of the object support as well as the background. This model uses a set of texture coefficients to represent the object and background inhomogeneities and a contour to represent the boundary of multiple connected or unconnected objects. The problem of determining the unknown contour and texture coefficients of the object and background medium is then formulated as a non-linear estimation problem. By designing a new, "tomographic flow", the resulting problem is recast into a curve evolution problem and an efficient algorithm based on level set techniques is developed. The performance of the curve evolution method is demonstrated using examples with noisy Radon transformed data and noisy ground penetrating radar data. The reconstruction results and computational cost are compared with those of conventional regularization methods. The results indicate that our curve evolution methods achieve improved shape reconstruction with reduced computation requirements.