Several techniques have been described in the literature in recent years for the reconstruction of a regular volume out of a series of ultrasound (US) slices with arbitrary orientations, typically scanned by means of US freehand systems. However, a systematic approach to such a problem is still missing. This paper focuses on proposing a theoretical framework for the 3-D US volume reconstruction problem. We introduce a statistical method for the construction and trimming of the sampling grid where the reconstruction will be carried out. The results using in vivo US data demonstrate that the computed reconstruction grid that encloses the region-of-interest (ROI) is smaller than those obtained from other reconstruction methods in those cases where the scanning trajectory deviates from a pure straight line. In addition, an adaptive Gaussian interpolation technique is studied and compared with well-known interpolation methods that have been applied to the reconstruction problem in the past. We find that the proposed method numerically outperforms former proposals in several control studies; subjective visual results also support this conclusion and highlight some potential deficiencies of methods previously proposed.