There are many imaging techniques that generate three-dimensional volume images today. With higher precision in the image acquisition equipment, storing and processing these images require increasing amount of data processing capacity. Traditionally, three-dimensional images are represented by cubic (or cuboid) picture elements on a cubic grid. The two-dimensional hexagonal grid has some advantages over the traditionally used square grid. For example, less samples are needed to get the same reconstruction quality, it is less rotational dependent, and each picture element has only one type of neighbor which simplifies many algorithms. The corresponding three-dimensional grids are the face-centered cubic (fcc) grid and the body-centered cubic (bcc) grids. In this thesis, image representations using non-standard grids is examined. The focus is on the fcc and bcc grids and tools for processing images on these grids, but distance functions and related algorithms (distance transforms and various representations of objects) are defined in a general framework allowing any point-lattice in any dimension. Formulas for point-to-point distance and conditions for metricity are given in the general case and parameter optimization is presented for the fcc and bcc grids. Some image acquisition and visualization techniques for the fcc and bcc grids are also presented. More theoretical results define distance functions for grids of arbitrary dimensions. Less samples are needed to represent images on non-standard grids. Thus, the huge amount of data generated by for example computerized tomography can be reduced by representating the images on non-standard grids such as the fcc or bcc grids. The thesis gives a tool-box that can be used to acquire, process, and visualize images on high-dimensional, non-standard grids.

Forelimbs of Day 11 and Day 12 embryonic mice were excised and cultured for 3 d in the presence of either 0.25 μg (8×10-7 M), 0.5 μg(1.7×10-6 M), or 1.0 μg (3.3×10-6 M) of all-rans retinoic acid (RA) per milliliter of culture medium. Cultured limbs were fixed, stained, and mounted whole on glass slides and evaluated with computerized optical image analysis for RA-induced effects on the area and shape of the total limb and individual bone anlagen. Relative effects of RA on total bone, soft tissue, long bone, and paw regions were also examined. With Day 11 forelimbs total bone area was increased by 10.5% by the low dose of RA. The increase was mostly in long bones and at the expense of soft tissue. Total bone area was increased 9.3% with Day 12 forelimbs. This increase was primarily in the paw. The high dose of RA decreased Day 11 forelimb area, primarily affecting long bones. Day 12 forelimbs were not significantly affected by the high dose of RA. Effects of the imtermediate dose were primarily limited to reduction in soft tissue area. Long bone:paw and soft tissue: bone ratios reflected these effects. The high dose produced a consistent rounding or shortening of Day 11 forelimb bones. On Day 12 0.5 μg/ml RA produced an inconsistent pattern of rounding of bone anlagen. Treatment with the high dose on Day 12 produced angular rather than rounded contours in many cases, as indicated by shape factor values closer to zero than obtained with controls. These data show that direct exposure to RA can affect both the size and shape of bone anlagen of the developing limb; the low dose enhances and the high dose depresses development. The results support previous studies which suggest that RA may play a critical role in the control of cell activities such as cell migration, proliferation, and cytodifferentiation in the development of the cartilaginous bone anlagen.