Interphase chromosomes are organized into discrete chromosome territories (CTs) that may occupy preferred sub-nuclear positions. While chromosome size and gene density appear to influence positioning, the biophysical mechanisms behind CT localization, especially the relationship between morphology and positioning, remain obscure. One reason for this has been the difficulty in imaging, segmenting, and analyzing structures with variable or imprecise boundaries. This prompted us to develop a novel approach, based on the two-dimensional (2D) wavelet-transform modulus maxima (WTMM) method, adapted to perform objective and rigorous CT segmentation from nuclear background. The wavelet transform acts as a mathematical microscope to characterize spatial image information over a continuous range of size scales. This multiresolution nature, combined with full objectivity of the formalism, makes it more accurate than intensity-based segmentation algorithms and more appropriate than manual intervention. Using the WTMM method in combination with numerical simulation models, we show that CTs have a highly nonspherical 3D morphology, that CT positioning is nonrandom, and favors heterologous CT groupings. We discuss potential relationships between morphology, positioning, chromosomal function, and instability.