List of Figures

• 1 An interpolated level set with simple topology.
• 2 Extra topological complications not directly implied by the data.
• 3 The possible arrangements of Highs on a face, with boundary contours.
• 4 Components of $\bf{X}_{f}^{\geq \tau}$ defined by edge-connected sets of Highs. Note this implies that the point p satisfies $f(p) < \tau$.
• 5 Inconsistent results when we interpolate components of $\bf{X}_{-f}^{> - \tau}$ defined by edge-connected sets of Highs. The point p satisfies $-f(p) \leq -\tau$ and thus $f(p) \geq \tau$, contradicting the previous figure.
• 6 Knit High above, Knit Low below: Initially there are two separate objects.
• 7 The two objects touch at the disambiguation value.
• 8 The two objects have merged.
• 9 The entire 4-hit face is engulfed.
• 10 Illustration of nomenclature for 4-hit face disambiguation.
• 11 The disambiguation point and value (for 2 dimensions) via linear interpolation across the face.
• 12 A Vertex Local Maximum: at thresholds above 20 there is no object, below 20 there is one object
• 13 A Local Minimum Critical Point: at thresholds above 10 there is a hole, below 10 the hole vanishes.
• 14 A Saddle Critical Point on a vertex: at thresholds above 20 there are two local objects, below 20 the two local objects merge.
• 15 A Critical 4-hit Face Saddle Critical Point.
• 16 A local maximum critical set, where region S has the constant value of 100.
• 17 A local minimum critical set, again where region S has the constant value of 100.
• 18 A Saddle Critical Set. Again region S has the constant value of 100.
• 19 Criticalities in the dataset (see table 1 for the critical values).
• 20 Zones of the 5 maxima shown Initially.
• 21 Zone of the criticality g has been added.
• 22 Zone of criticality h has been added
• 23 Zones of saddles i and j. At this point the objects have merged into one object with two holes. The two remaining criticalities are the minima k and l. Zone of k will include all remaining points with value greater than 50 and zone of l will include all the remaining points.
• 24 Criticality graph associated with the dataset.
• 25 Spinning one triangle in a cube face.
• 26 Illustration of a simple corner intersection of a surface with a cube. Note that each of the three faces have two hits. Each pair of hits in a face is connected back to the Articulation point. The Articulation point (center of moment of the three grouped hits, corresponding to one surface and one criticality beyond) is used to knit three triangles.
• 27 Illustration of a surface intersecting the waist of a cube, passing through the center of the cube, instead of passing through a corner.
• 28 Two examples of two separate surfaces passing through the corners of a cube. The left cube has a surface cutting through the two front faces of the cube. The right cube has a surface passing through the front surface, and another, possibly separate surface passing through the back face of the cube.
• 29 Triangle edge from AP to h is shared by two triangles, defined by the hits adjacent to h in each of the cube faces that shares the hit. These two hits are g and g'.
• 30 The triangle edge connecting two adjacent hits is shared by the two triangles defined by an articulation point in each of the two cubes that share the face on which the hits occur.
• 31 Overhead view of the eight edges emanating from a central hit point, into the 4 shared cubes. Each X marks a hit.
• 32 Cyclical enumeration of edges from AP to hits on the cube faces.
• Artist's iIllustration of critical and non-critical 4-hit faces. The 3 cube faces represent two adjacent cubes with the shared face a 4-hit face. In each example the face of the top cube is outside the level set, and we show the 3 faces above and below the disambiguation value (D.V.). The solid figure represents the level set object(s), up to topological equivalence, through the bottom cube. This is not a critical 4-hit face.
• 34 The shared face is not a critical 4-hit face, because the two Highs are already part of the component in the bottom cube above the D.V.
• 35 The shared face is not a critical 4-hit face, because the two Highs are already part of the component in the bottom cube above the D.V.
• 36 The shared face is a critical 4-hit face, because the two Highs are not part of the component in the bottom cube above the D.V.
• 37 The shared face is a critical 4-hit face, because the two Highs are not part of the component in the bottom cube above the D.V.
• 38 The shared face is a critical 4-hit face, because the two Highs are not part of the component in the bottom cube above the D.V.
• 39 The shared face is a critical 4-hit face, because the two Highs are not part of the component in the bottom cube above the D.V.