1. Motivation Skeleton needed for mesh deformation:
Manual construction: tedious
Automatic construction: quality problems
Idea: ad- hoc by sketching
Input:
Stroke(s) on mesh surface
Output:
3D skeleton
3D region of interest

2. Related Work Control structures: FFD lattice [SP86] Wires [SF98]
Skeleton [CGC*02, YBS03]
Ad-hoc control structures:
Scalar field [HQ03]
Silhouette [NSAC05]
Strokes on surface:
Small deformations [OSSSJ05, Buj06, CJ006]
Large deformations [ZHS*05, KG05]

3. Kho & Garland [KG05] Input: Single stroke on mesh surface Output:
2D control structure – the stroke itself
Two cutting planes delimiting region of interest
Limitations:
No 3D skeleton
No 3D region of interest
Cutting planes always orthogonal to sketch plane
Complex deformations require multiple steps

4. Our Approach 3D skeleton: By projecting from surface inward
3D region of interest:
Described by cuts orthogonal to the skeleton
Chicken-and-egg problem:
Skeleton: projection into region of interest
Region of interest: built around skeleton
Our approach:
Build cutting planes for points along the stroke
Find consistent orientations by optimization
Build skeleton by projecting toward cut centers

5. Overview

6. 1: Stroke on Surface

7. 2: Cutting Planes

8. 3: Cuts

9. 4: Optimization

10. 5: Projection

11. Cutting Plane Initially orthogonal to the stroke
Can be rotated around:
Surface normal: α
Binormal in tangential plane: β

12. Iterative Local Optimization
Process:
for each plane:
set αi = 0 and βi = 0
for each iteration:
calculate αi,j and βi,j
αi = αi + Σ wj·αi,j
βi = βi + Σ wj·βi,j
Optimization criteria:
Neighboring planes should be nearly parallel
Cuts should not intersect each other
Planes should cut “naturally” across the mesh

13. Global Adjustment Local criteria may rotate planes inconsistently:
Global consistency is very difficult to assess
Our method – periodic sweeps:
αi+1 = αi and βi+1 = βi
Aligns each plane with its predecessor
Successive sweeps propagate new orientations

14. Dynamic Programming Graph: Vertex: candidate plane
Edge: measure of consistency between planes
Least cost path:
Optimal plane arrangement
Found by dynamic programming (“backtracking”)
Edge length weighted sum of:
Difference in angle
Difference in cut circumference
Difference in form factor
β0,0
β0,1
β0,2
β1,0
β1,1
β1,2
β2,0
β2,1
β2,2

15. Challenges Weights: Manual tuning necessary
No universally applicable settings
Iterative optimization:
Rotation must be artificially constrained
Without global adjustment:
Inconsistent cutting plane arrangements
With global adjustment:
Alignment dominated by first plane
Dynamic programming:
Only one angle can be optimized due to cost

16. Alternative: Two Strokes
User specifies first and last cutting planes
Two interpretations possible:
Regression plane: plane fully specified by user
Regression line: rotation axis specified by user
Geodesic between planes projected as skeleton

17. Summary Proposed 3D control structure construction:
1. Region of interest: build and align cutting planes
2. Skeleton: project from surface inward
Results:
Single stroke:
Weights require tuning
Global assessment difficult
Dynamic programming expensive
Two strokes:
Promising approach
Global shape delimited by given planes
Intermediary planes need optimization

18. Future Work Refine the two-stroke approach: Stroke interpretation:
Regression plane, regression line, ...
Intermediary planes:
Reduce/eliminate optimization weights
Integrate with mesh deformation method:
Deform skeleton to match a target shape
Use 3D region of interest to prevent self- intersections