Inversion of limited data is common in many areas of NDE such as X-ray Computed Tomography (CT), Ultrasonic and eddy current flaw characterization and imaging;In many applications, it is common to have a bias toward a solution with minimum (L[superscript]2)[superscript]2 norm without any physical justification. When it is a priori known that objects are compact as, say, with cracks and voids, by choosing "Minimum Support" functional instead of the minimum (L[superscript]2)[superscript]2 norm, an image can be obtained that is equally in agreement with the available data, while it is more consistent with what is most probably seen in the real world;We have utilized a minimum support functional to find a solution with the smallest volume. This inversion algorithm is most successful in reconstructing objects that are compact like voids and cracks. To verify this idea, we first performed a variational nonlinear inversion of acoustic backscatter data using minimum support objective function. A full nonlinear forward model was used to accurately study the effectiveness of the minimized support inversion without error due to the linear (Born) approximation. After successful inversions using a full nonlinear forward model, a linearized acoustic inversion was developed to increase speed and efficiency in imaging process. The results indicate that by using minimum support functional, we can accurately size and characterize voids and/or cracks which otherwise might be uncharacterizable;An extremely important feature of support minimized inversion is its ability to compensate for unknown absolute phase (zero-of-time). Zero-of-time ambiguity is a serious problem in the inversion of the pulse-echo data. The minimum support inversion was successfully used for the inversion of acoustic backscatter data due to compact scatterers without the knowledge of the zero-of-time;The main drawback to this type of inversion is its computer intensiveness. In order to make this type of constrained inversion available for common use, work needs to be performed in three areas: (1) exploitation of state-of-the-art parallel computation, (2) improvement of theoretical formulation of the scattering process for better computation efficiency, and (3) development of better methods for guiding the non-linear inversion. (Abstract shortened by UMI.)