ady ecker, kiriakos n. kutulakos and allan d. jepson ...adyecker/download/poster.pdfady ecker,...
TRANSCRIPT
For curves arranged around a principle plane,
the shape might be a singular vector of ππ ,
ππ― 2 = π πππ(ππ) + πππ(ππ)
Shape from planar curves: a linear escape from FlatlandAdy Ecker, Kiriakos N. Kutulakos and Allan D. JepsonUniversity of Toronto
βHumans can perceive 3D from parallel planar cross sections
βDo not assume fixed gaps between the planes. Not standard shape from texture
βWe demonstrate reconstructions in special cases
Local planarityβReduce to the intersecting
case by assuming virtual planar facets
Parallel curves References
We revisit the problem of recovering 3D shape from a single photo containing planar curves. Previous work [1,2,3] did not explicitly explore the space of flat solutions and its orthogonal complement
β For intersecting planar curves, we study the space of solutions and derive a stable linear method
β For parallel planar curves, we demonstrate special cases where the shape is similarly solved
βOur work unifies relevant literature on shape from contour, single view modeling and structured light
Introduction
Intersecting curvesConsider planar curves
π§π π₯,π¦ = πππ₯ + πππ¦ + ππ
At intersection points
π§π π₯ππ ,π¦ππ β π§π π₯ππ ,π¦ππ =
ππ β ππ π₯ππ + ππ β ππ π¦ππ + ππ β ππ = 0
The homogenous linear system in vector form
ππ― = 0
π― = π1,β¦ ,ππ , π1,β¦ , ππ ,π1,β¦ ,ππ π
Let π be a matrix so that ππ― 2 is the linear regression residue of
fitting a plane to a sample of 3D points on the curves
image 3D planar curves
π₯ππ ,π¦ππ
βBreaks in presence of straight lines
βMay collapse to a nearly-flat solution
Simple linear method
argminπ―
ππ― s.t. π― = 1,π― β₯ Span{π―1, π―2, π―3}
Our method
argmin π―
ππ― s.t. ππ― = 1, π― β₯ ππ’ππ(π)
π― = π+π°, where π° is the last right singular vector of ππ+
Similar linear formulation for perspective projection
β Singular values are independent of the focal length
Ambiguity resolution: pick a vector orthogonal to the trivial subspace
β For a large number of random planes, the relative magnitude of the trivial component is likely to be small
Systems with lines
have additional
trivial solutions
Space of solutions ππ― = 0
Trivial subspace ππ― = 0 Non-trivial solutions ππ― β 0
π―1 = 1π ,02π
π
π
π―2 = 0π ,1π ,0π
π
π
π―3 = 02π ,1π
π
π
Basic trivial
subspace
(GBR ambiguity)
True surface (1D ambiguity)
Additional non-trivial solutions
Under-constrained Over-constrained
Straightcurve
K. Sugihara. Machine Interpretation of Line Drawings. MIT press, 1986
C. Rothwell and J. Stern. Understanding the shape properties of trihedral polyhedra. Technical Report 2661, INRIA, 1995
J.-Y. Bouguet, M. Weber, and P. Perona. What do planar shadows tell us about scene geometry? In Proc. CVPRβ99, pages 514β520, 1999
[1]
[2]
[3]
parallel curves sampled points
image intersecting 2D curves computed 3D surface
Global planarityβ
sample points and virtual facets
solve the linear system
computed 3D surface
1
π ππ , ππ π1,β¦ , ππ , π1,β¦ , ππ