Single figure coordinate system

Medial geometry (Blum (1967); Siddiqi & Pizer (2008)) describes 3D objects in terms of a skeletal surface or axis, which lies midway between opposing surfaces of the object. Depending on the type of object, slab-shaped or tube-shaped, different parameterization

techniques are used for medial geometry:

1. Slab parameterization: In this parameterization medial geometry describes 3D objects in terms of a skeletal surface, a 2D curved sheet, and a set of spokes extending to the object boundary from both sides of the skeletal surface. The continuous medial manifold,M, of a 3D slabular object is a sheet of medial atoms parameterized by (u, v), the coordinates along the medial manifold, with u and

v taking the atom index numbers for sample positions along and across M. The single entity M(u, v) is called a medial atom. For slabular objects the medial atom is a modeling primitive that represents a track through the object interior (Fig. 2.14-left). Each medial atom has a hub, which is a point on the medial sheet

p(u, v). From each hub two or three spokes extend from the medial sheet to the corresponding boundary points. To represent a hub, two spokes, and the common length of those spokes, the medial atom is defined as a 4-tuple {p, r, U+1, U−1} where p is the position vector, r is spoke length, and U+1, U−1 are the two spoke orientations as two unit vectors (Figure 2.14-left). The medial atoms on the edge of the medial sheet correspond to crests of the object boundary. Such an end atom adds a bisector spoke U0 and its length r0 to the parameterization (Figure 2.14-

right). In this parameterization moving in the along-object (u) direction refers to moving along the longer direction on the medial surface on both medial sides of spokes. Moving in the across-and-around-object (v) direction refers to moving along the shorter direction on the medial surface on one medial side and turning onto the other side at the crest (Figure 2.15).

2. Quasi-tube parameterization: In this parameterization medial geometry describes 3D objects in terms of a skeletal curve and a set of spokes extending to the object boundary from the sides of the skeletal curve. The continuous medial manifold,M, of a 3D tube-shaped object is a curve of wheel-like medial atoms parameterized by u, the coordinate along the medial curve. In this parameterization v moves

Figure 2.14: (Left) an internal medial atom; (right) a crest atom.

Figure 2.15: A single figure slabular m-rep for the scm and the object boundary implied by it.

cyclically around the wheel of spokes forming a cross section of the quasi-tube. The single entity M(u) is called a medial atom. For tubular objects the medial atom is a modeling primitive that represents a plane of tracks through the object interior. Each medial atom has a hub, which is a point on the medial sheet p(u). From each hub spokes extend from the medial sheet to the corresponding boundary points. To represent a hub, spokes, and the common length of those spokes for a tube (quasi-tube in which every wheel of spokes is circular), the medial atom is defined as a 4-tuple{p, r, U0} wherepis the position vector, r is spoke length,U0

is the orientation orthogonal to the wheel. For a quasi-tube each spoke in a wheel must have its own r value, and these values must be chosen such that each spoke ends in a common plane (Figure 2.16-right). In this parameterization moving in the along-object (u) direction refers to moving along the medial curve for all wheel

spokes, and moving in the around-object (v) direction refers to rotating around the wheels (Figure 2.17).

Figure 2.16: (Left) an internal medial atom for a tube; (right-top) a quasi-tube atom with spokes; (right-bottom) cut-away section of the surface.

Figure 2.17: A single figure quasi-tubular m-rep for the pharynx and the object boundary implied by it.

The final coordinate, measuring distance along the spoke directions from the implied boundary is denoted by τ. It stores the fraction of the spoke from the medial sheet to the boundary. This coordinate can be used for distinguishing the inside regions of the figure from the outside. That is, τ <1 in the interior of the figure,τ > 1 in the exterior of the figure, and τ = 1 on the boundary.

Given an m-rep figure, a smooth object surface is generated to interpolate the bound- ary positions and normals implied by the atom spokes using either Thall’s subdivision method (Thall (2002)) or Han’s spoke interpolation method (Han (2008)) (the former was used in this dissertation). If (u, v) parametrizes the spokes emanating from the

medial sheet or curve, the implied boundary is parametrized by (u, v). Thereby, the coordinates on the medial sheet can be transferred to the boundary of the model. In the slab parameterization there are two corresponding points (u, v) on the boundary for every hub p(u, v) on the medial sheet. In the quasi-tube parameterization there are coplanarv points on the boundary for each hubp(u) on the medial curve. For two differ- ent instances of a model the correspondence is defined by the same medial coordinates for this model, which is also helpful to indicate the similar points in the instances of the model. In this way, m-reps provide positional, metric, and orientation correspondences (Fig. 2.18). In this dissertation correspondence is very important since we want consis- tent orientation fields and material transition fields for different instances of the same model.

Figure 2.18: Medial-based correspondences between a figure and a deformed figure for boundary positions (left), and for interior and exterior positions to the boundary (mid- dle). Boundary points on a deforming kidney (right).

Chapter 3

Texture Metamorphosis

In anatomical illustrations different regions of anatomical models are visualized using different material textures. The transition between these regions can be illustrated with textures that are perceptually between the two material textures.

To obtain this effect in the model-guided texture synthesis (MGTS) framework, region-specific exemplar textures along with the transition textures between them are needed as input. In this dissertation texture metamorphosis is used to create these transition textures by morphing one region-specific texture into another region-specific texture.

This chapter presents a technique that is accepted to the Computer Graphics Forum in 2011 as a journal publication (Kabul et al. (2011)).