Numerical geometry of non-rigid shapes Shortest path problems Alexander Bronstein, Michael Bronstein, Ron Kimmel © 2007 All rights reserved

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Shortest paths in graphs Object is sampled at vertices Approximated as undirected graph Edges represent adjacent samples Length function measures local length that can be approximated as Path in graph between two vertices and given by Path length – sum of lengths of all edges

Shortest paths in graphs Approximate the intrinsic metric as the shortest path length in graph Alternative formulation: given a source point , compute the distance function Bellman’s principle of optimality: if is the shortest path between and , and is a point on it, then the sub-paths and are the shortest paths between and , and and , respectively.

Shortest paths in graphs Consequence: there exists some such that How to select such a point ? Since has to be the shortest distance, Problem of computing is reduced to smaller sub-problems of computing Dynamic programming

Dijkstra’s algorithm Dynamic programming successive approximation procedure Initially, set ; for the rest of the vertices, set Mark all vertices as unprocessed While there are still unprocessed vertices Select with the smallest Update all adjacent points Mark as processed Return the distance function

Dijkstra’s algorithm Every vertex is processed only once Update step propagates distance to adjacent vertices If the previously computed distance is shorter, value is not updated Typically, there are vertices in the graph Extraction of the minimum distance can be done in using a heap Total algorithm complexity

Metrication error “True” distance is Distance measured in graph Error persists, no matter how the sampling is refined

Metrication error No flaw in Dijkstra’s algorithm The graph induced the metric, inconsistent with the Euclidean metric Problem can be resolved by changing the sampling So far the path was restricted to graph edges Represent the surface as a triangle mesh Allow the path to pass on mesh faces Fast marching – a “continuous Dijkstra” algorithm Numerical solution of the eikonal equation

Fast marching Dijkstra Path restricted to graph edges Adjacent vertices are updated A vertex is updated from another vertex Fast marching Path can pass on mesh edges Adjacent triangles are updated A vertex is updated from two vertices within a triangle

Fast marching update step Given a triangle with vertices , , Given distances and We need to compute Model planar source described by Wavefront propagates from the source Hits at time

Fast marching update step From triangle geometry and , find the source parameters Propagation direction: (1 DOF) Offset: (1 DOF) Solve Quadratic equation with two solutions corresponding to Smallest solution is inconsistent and is discarded Compute

Consistency & monotonicity Consistency condition: Wavefront hits and before hitting Monotonicity condition: Increase of or increases

Geometric interpretation Consistent & Monotone Monotone Consistent Consistency condition: update direction must form acute angles with normals to triangle edges Monotonicity condition: update direction must come from within the triangle

Monotonicity Monotonicity: update must come from within the triangle If is not within the triangle, force it to lie inside The update direction will coincide with one of the edges Dijkstra-type update

Consistency: acute triangle When triangle is acute, both conditions hold for any direction

Consistency: obtuse triangle When triangle is obtuse, some consistency condition is violated for some directions

Fast marching on obtuse meshes Inconsistent solution if the mesh contains obtuse triangles Remeshing is costly Solution: split obtuse triangles by adding virtual connections to non-adjacent vertices Done as a pre-processing step in

Fast marching algorithm Set and mark it as processed Mark all other vertices as unprocessed and set While there are still unprocessed vertices Select with the smallest Update all triangles sharing Mark as processed Return the distance function

Fast marching algorithm Geodesic distances Minimal geodesics Voronoi tessellation

Fast marching on parametric surfaces Object is given as a parametric surface Parametrization domain is sampled on a regular grid (geometry image) Fast marching can be performed entirely in the parametrization domain Update step accounts for the surface first fundamental form to measure geodesic distances on the surface Surface itself is not required, knowledge of the first fundamental form and grid connectivity is sufficient Obtuse triangles are split by adding virtual connections to non-adjacent grid points

Heap-based grid update Fast marching on triangular meshes and geometric images both use Dijstra-type heap-based grid update Update order is unknown and data-dependent Inefficient use of memory system and cache Inherently sequential algorithm – next update depends on previous one

Raster scan fast marching Update the grid in a raster scan order Dates back to Danielsson’s algorithm In Euclidean case, geodesics are straight lines Each raster scan covers ¼ of the possible directions of the geodesics Four raster scans covering four quadrants are sufficient to compute a Euclidean distance map

Raster scan fast marching 1 iteration 2 iterations 3 iterations Generally, geodesics are curved in parametrization domain Raster scans have to be repeated to produce a convergent solution Number of iterations depends on geometry and parametrization Practically, few iterations are required

Raster scan fast marching What we lost: No more a one-pass algorithm Computational complexity is data-dependent What we gained: Coherent memory access, efficient use of cache No heap, each iteration is Raster scans can be parallelized

Marching even faster All updates performed on a diagonal are independent Can be computed simultaneously Uneven load

Marching even faster Rotate scan directions by 45 degrees All updates performed along a row or column can be parallelized Balanced load Suitable for SIMD architecture and GPUs x100 speedup with comparable accuracy

Conclusions so far… We have all numerical tools required to approximate the intrinsic geometry of objects Graph representation may introduce metrication error Intrinsic metric approximated using Dijkstra’s algorithm may be inconsistent Fast marching is a consistent numerical tool for measuring geodesic distances Fast marching can be parallelized