Image Morphing
slides mostly from Alexei Efros, Berkeley
CS3335 Visual computing Image Morphing
• •
Cross-dissolving images
Morphing images
• Delaunay triangulation
• piece-wise affine transforms/warps
• Combining cross-dissolve (intensity interpolation)
with gradual shape deformation (warp interpolation)
Morphing = Object Averaging
The aim is to find “an average” between two objects
• Not an average of two images of objects…
• …but an image of the average object!
• How can we make a smooth transition in time?
– Do a “weighted average” over time t
How do we know what the average object looks like?
• We haven’t a clue!
• But we can often fake something reasonable – Usually required user/artist input
Interpolating Images
v=Q-P
P
P + 0.5v
= P+0.5(Q–P) = 0.5P+0.5Q (interpolation)
P + 1.5v
= P+1.5(Q–P) = -0.5P+1.5Q (extrapolation)
Q
P and Q can be anything:
• points on a plane (2D) or in space (3D)
• Colors in RGB or HSV (3D)
• Whole images (m-by-n D)… etc.
Idea #1: Cross-Dissolve
Interpolate whole images:
Imagehalfway = (1-t)*Image1 + t*image2
This is called cross-dissolve in film industry
But what is the images are not aligned?
Idea #2: Align, then cross-disolve
Align first, then cross-dissolve
• Alignment using global warp – picture still valid
Dog Averaging
What to do?
• Cross-dissolve doesn’t work
• Global alignment doesn’t work
– Cannot be done with a global transformation (e.g. affine)
• Any ideas?
Feature matching!
• Nose to nose, tail to tail, etc.
• This is a local (non-parametric) warp
Idea #3: Local warp, then cross-dissolve
for every t,
Morphing procedure:
1. Find the average shape (the “mean dog”) • local warping
2. Find the average color
• Cross-dissolve the warped images
Local (non-parametric) Image Warping
Need to specify a more detailed warp function
• Global warps were functions of a few (2,4,8) parameters
• Non-parametric warps u(x,y) and v(x,y) can be defined independently for every single location x,y!
• Once we know vector field u,v we can easily warp each pixel (use backward warping with interpolation)
Image Warping – non-parametric
Move control points to specify a spline warp Spline produces a smooth vector field
Warp specification – dense How can we specify the warp?
Specify corresponding spline control points
• interpolate to a complete warping function
But we want to specify only a few points, not a grid
Warp specification – sparse How can we specify the warp?
Specify corresponding points
• interpolate to a complete warping function • Howdowedoit?
How do we go from feature points to pixels?
Triangular Mesh
1. Input correspondences at key feature points
2. Define a triangular mesh over the points
• Same mesh in both images!
• Now we have triangle-to-triangle correspondences
3. Warp each triangle separately from source to destination
• How do we warp a triangle?
• 3 points = affine warp!
• Just like texture mapping
Triangulations
A triangulation of set of points in the plane is a partition of the convex hull to triangles whose vertices are the points, and do not contain other points.
There are an exponential number of triangulations of a point set
Any triangulation of a given set of n points will have the same number of triangles (2n-2-k where k is the number of points on the boundary of the convex hull) and the same number of edges (3n-3-k).
– p.185, “Computational Geometry” by M. Berg et al.
An O(n3) Triangulation Algorithm
Repeat until impossible:
• Select two sites.
• If the edge connecting them does not intersect previous edges, keep it.
“Quality” Triangulations
Let (T) = (1, 2 ,.., 3t) be the vector of angles in the
triangulation T in increasing order.
A triangulation T1 will be “better” than T2 if (T1) > (T2)
lexicographically.
The Delaunay triangulation is the “best” • Maximizes smallest angles
good bad
Improving a Triangulation
In any convex quadrangle, an edge flip is possible. If this flip improves the triangulation locally, it also improves the global triangulation.
If an edge flip improves the triangulation, the first edge is called illegal.
Illegal Edges
Lemma: An edge pq is illegal iff one of its opposite vertices is inside the circle defined by the other three vertices.
Proof: By Thales’ theorem.
p
q
Theorem: A Delaunay triangulation does not contain illegal edges. Corollary: A triangle is Delaunay iff the circle through its vertices is
empty of other sites.
Corollary: The Delaunay triangulation is not unique if more than three sites are co-circular.
Naïve Delaunay Algorithm
Start with an arbitrary triangulation. Flip any illegal edge until no more exist.
Could take a long time to terminate.
Delaunay Triangulation by Duality
General position assumption: There are no four co-circular points.
Draw the dual to the Voronoi diagram by connecting each two neighboring sites in the Voronoi diagram.
Corollary: The DT may be constructed in O(nlogn) time.
This is what Matlab’s delaunay function uses.
both scientists worked
on mathematical
crystallography
Boris Delaunay 1890 – 1980
Georgy Voronoy 1868-1908
Useful properties of Voronoi cells Theorem: For n≥3, the number of edges in the Voronoi
diagram of a set of n points in the plane is at most 3n-6. – p.150, “Computational Geometry” by M. Berg et al.
Corollary: the average number of facets per Voronoi cell does not exceed 6.
– note: each edge is shared by 2 cells
Image Morphing
We know how to warp one image into the other, but how do we create a morphing sequence?
1. Create an intermediate shape (by interpolation)
2. Warp both images towards it
3. Cross-dissolve the colors in the newly warped images
Warp interpolation
How do we create an intermediate warp at time t?
• Assume t = [0,1]
• Simple linear interpolation of each feature pair
• (1-t)*p1+t*p0 for corresponding features p0 and p1
Other Issues
Beware of folding
• You are probably trying to do something 3D-ish
Morphing can be generalized into 3D • If you have 3D data, that is!
Extrapolation can sometimes produce interesting effects • Caricatures
Dynamic Scene