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CMU CS 15463 - Image Morphing

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Image MorphingMorphing = Object AveragingAveraging PointsIdea #1: Cross-DissolveIdea #2: Align, then cross-disolveDog AveragingIdea #3: Local warp, then cross-dissolveLocal (non-parametric) Image WarpingImage Warping – non-parametricWarp specificationSlide 11Triangular MeshTriangulationsAn O(n3) Triangulation Algorithm“Quality” TriangulationsImproving a TriangulationIllegal EdgesO(n4) Delaunay Triangulation AlgorithmNaïve Delaunay AlgorithmDelaunay Triangulation by DualitySlide 21Beier&Neely (SIGGRAPH 1992)Algorithm (single line-pair)Multiple LinesResulting warp (complex!)Full AlgorithmResultsDynamic SceneSlide 29Warp interpolationOther IssuesVideo MatchingImage Morphing15-463: Rendering and Image ProcessingAlexei EfrosMorphing = Object AveragingThe 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 tHow 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 inputPQv = Q - PP + 0.5v= P + 0.5(Q – P)= 0.5P + 0.5 QP + 1.5v= P + 1.5(Q – P)= -0.5P + 1.5 Q(extrapolation)Linear Interpolation(Affine Combination):New point aP + bQ,defined only when a+b = 1So aP+bQ = aP+(1-a)QAveraging PointsP 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.What’s the averageof P and Q?Idea #1: Cross-DissolveInterpolate whole images:Imagehalfway = t*Image1 + (1-t)*image2This is called cross-dissolve in film industry•Note similarity to alpha blending!But what is the images are not aligned?Idea #2: Align, then cross-disolveAlign first, then cross-dissolve•Alignment using global warp – picture still validDog AveragingWhat to do?•Cross-dissolve doesn’t work•Global alignment doesn’t work–Cannot be done with a global transformation (e.g. prospective)•Any ideas?Feature matching!•Nose to nose, tail to tail, etc.–This is a local (non-parametric) warpIdea #3: Local warp, then cross-dissolveMorphing procedure: for every t,1. Find the average shape (the “mean dog”)•local warping2. Find the average color•Cross-dissolve the warped imagesLocal (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)Optical flow is just such a vector field•Will it work for these dogs?•Probably not… Need user control.Image Warping – non-parametricMove control points to specify a spline warpSpline produces a smooth vector fieldWarp specificationHow can we specify the warp?3. Specify corresponding spline control points•interpolate to a complete warping functionBut we want to specify only a few points, not a gridWarp specificationHow can we specify the warp?1. Specify corresponding points•interpolate to a complete warping function•How do we do it?Triangular Mesh1. Input correspondences at key feature points2. Define a triangular mesh over the points•Same mesh in both images!•Now we have triangle-to-triangle correspondences3. Warp each triangle separately from source to destination•How do we warp a triangle?•3 points = affine warp!•Just like texture mappingTriangulationsA 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.An O(n3) Triangulation AlgorithmRepeat until impossible:•Select two sites.•If the edge connecting them does not intersect previous edges, keep it.“Quality” TriangulationsLet (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 anglesgood badImproving a TriangulationIn 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 EdgesLemma: 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.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.pqO(n4) Delaunay Triangulation AlgorithmRepeat until impossible:•Select a triple of sites.•If the circle through them does not contain other sites, keep the triangle whose vertices are the triple.Naïve Delaunay AlgorithmStart with an arbitrary triangulation. Flip any illegal edge until no more exist.Requires proof that there are no local minima.Could take a long time to terminate.Delaunay Triangulation by DualityGeneral 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.Warp specificationHow can we specify the warp?2. Specify corresponding vectors•interpolate to a complete warping function•The Beier & Neely AlgorithmBeier&Neely (SIGGRAPH 1992)Single line-pair PQ to P’Q’:Algorithm (single line-pair)For each X in the destination image:1. Find the corresponding u,v2. Find X’ in the source image for that u,v3. destinationImage(X) = sourceImage(X’)Examples:Multiple LinesLength = length of the line segment, dist = distance to line segmenta, p, b – constants. What do they do?Resulting warp (complex!)Full AlgorithmResultsDynamic SceneImage MorphingWe know how to warp one image into the other, but how do we create a morphing sequence?1. Create an intermediate warping field (by interpolation)2. Warp both images towards it3. Cross-dissolve the colors in the newly warped imagesWarp interpolationHow do we create an intermediate warp at time t?For optical flow?•Easy. Interpolate each flow vector•That’s how


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CMU CS 15463 - Image Morphing

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