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Continuous Morphology and Distance Maps Ron Kimmel www.cs.technion.ac.il/~ron Computer Science Department Technion-Israel Institute of Technology Geometric Image Processing Lab
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Given a closed planar curve Define the distance map Distance map
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Distance Map Properties Almost everywhere The level sets of, given by are the offsets of C
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Distance Map Properties By Huygens principle a level sets of, is given by the envelope of all disks of radius c centered on the curve C. The new shape is also known as `dilation’ with a circular `structuring element’ of the shape.
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Distance Map Properties The distance map represents the set, generated by the curve evolution with the right `entropy condition’
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Distance Map Properties The vector is pointing to the closest point on the zero set
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How to Compute? Accuracy/Efficiency Q1: How to compute the distance from a single `source point’? Given T(k,l)=0, find Solution: So ???
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How to Compute? Q2: How to compute the distance from two `source points’? Given T(k,l)=0, find Solution: Complexity:
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How to Compute? Q3: can it be computed in O(N) ? Solution: Danielson algorithm. 4 scans algorithm with alternating directions (up/down left/right) Ask your 4 neighbors their coordinate offset to the closest detected source point. Compute your offset to that point and decide if you change your choice of closest source point Initial offset is
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Danielson algorithm O(N)
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Alternative solution We can also use a numeric scheme to solve the ‘eikonal’ equation Initialize all source points and all non-source points Solve the quadratic equation for given by the upwind monotone approximation of the eikonal equation Again, use the 4 scans with alternating directions
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3D/4D/…nD All these methods can be extended to higher dimensions. 1D->2 2D->4 3D->8 4D->16
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Continuous Morphology Structuring element
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? Morphology and dual spaces
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Gray Scale Erosion and Dilation Level set by level set: E.g. Cylinder (level set circle)
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1.Segment at local curvature maxima 2.Compute distance map from each segment 3.Find intersection sets of the distance functions 4.Prune the tails Skeletons and level sets
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