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APES: Ghrist, Rossignac, Szymczak, Turk1 NSF CARGO DMS Georgia Tech, May 2004 Morphological Simplification Jason Williams and Jarek Rossignac GVU Center, IRIS Cluster, and College of Computing Georgia Institute of Technology

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APES: Ghrist, Rossignac, Szymczak, Turk2 NSF CARGO DMS Georgia Tech, May 2004 Overarching objective SIMPLIFICATION Understand what it means to simplify a shape or behavior Develop explicit mathematical formalisms independently of any particular domain or representation not stated as the result of some algorithmic process Propose practical implementations MULTI-SCALE ANALYSIS Explore the possibility of analyzing the evolution of a shape or behavior, as it is increasingly simplified, so as to understand its morphological structure and identify/measure its features identify features of interior, boundary, exterior assess their resilience to simplification

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APES: Ghrist, Rossignac, Szymczak, Turk3 NSF CARGO DMS Georgia Tech, May 2004 Complexity measures for 2D shapes Many measures of complexity are useful in different contexts: Visibility: Convex, star… number of guards needed Stabbing: Number of intersections with “random” ray Wiggles: energy in high frequency Fourier coefficients Algebraic: Polynomial degree of bounding curves Fidelity: accuracy required (Hausdorff, area preservation) Processing: Number of bounding elements Transmission: compressed file size Fractal: Fractal dimension Kolmogorov: Length of data and program We focus primarily on morphological ones: Sharpness: curvature statistics Smallest feature size: distance between non adjacent parts of boundary Topological: Number of components and holes Non-roundness: perimeter 2 /Area

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APES: Ghrist, Rossignac, Szymczak, Turk4 NSF CARGO DMS Georgia Tech, May 2004 What do we simplify and how much? Simplification replaces a shape by a simpler one What do we mean by “simpler”? Informally, we want to –Remove details Reduce sharpness and wiggles Eliminate small components and holes Hence increase the smallest feature size –Shorten (tighten) the perimeter While minimizing local changes in density –Ratio of interior points per square unit How much do we want to simplify –We want to be able to use a geometric measure, r, to specify which of the details should be simplified and how –What does the measure mean? What is the size of a detail?

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APES: Ghrist, Rossignac, Szymczak, Turk5 NSF CARGO DMS Georgia Tech, May 2004 Restricting simplification to a tolerance zone We want to restrict all changes to the envelop: tight zone around the boundary bS of the shape S –the yellow fat curve, is the offset (bS) r f bS by r We further restrict all changes to the Mortar –the green region, is the mortar M r (bS)=(bS r) r of S We explain why in the next few slides Envelop Mortar

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APES: Ghrist, Rossignac, Szymczak, Turk6 NSF CARGO DMS Georgia Tech, May 2004 We need a vocabulary To discuss and to measure simplicity, we need precise terms We will use three different measures –Smothness (value defined using Differential Geometry) –Regularity (value defined using Morphology) –Mortar (area near the details defined using Morphology) We want to be able to: –Measure smoothness at every point of the boundary of the shape –Measure regularity at every point in space –Measure the global regularity and smoothness of the whole shape –Define and compute the Mortar for the desired tolerance –Simplify the shape in the Mortar: increasing its regularity & smoothness A boundary point may be r-smooth or not, r-regular or not

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APES: Ghrist, Rossignac, Szymczak, Turk7 NSF CARGO DMS Georgia Tech, May 2004 A shape S is r-smooth if the curvature of every point B in its boundary bS exceeds r How to check for r-smoothness at B? –For C 2 curves: compare radius of curvature to r –For polygons: estimate radius of curvature R=v 2 /(a v) z, where v=AC/2 and a=BC–AB A point may be r-smooth, but not r-regular r-smoothness r-smooth not r-smooth r: A B C

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APES: Ghrist, Rossignac, Szymczak, Turk8 NSF CARGO DMS Georgia Tech, May 2004 r-regularity A shape S is r-regular if S=F r (S)=R r (S) –F r (S) = S r r, r-Fillet (closing) = area not reachable by r-disks out of S –R r (S)= S r r, r-Rounding (opening) = area reachable by r-disks in S –Each point of bS can be approached by a disk(r) in S and by one out of S Original L is not r-regular Removing the red and adding the green makes it r-regular This shape is r-smooth, but not r-regular

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APES: Ghrist, Rossignac, Szymczak, Turk9 NSF CARGO DMS Georgia Tech, May 2004 Boundary is resilient to thickening by r – bS can be recovered from its rendering as a curve of thickness 2r One-to-one mapping from boundary to its two offsets by r –The boundary of S r (resp. S r) may be obtained by offsetting each point of bS along the outward (resp. inward) normal. No need to trim. r-regularity implies r-smoothness Properties of r-regular shapes

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APES: Ghrist, Rossignac, Szymczak, Turk10 NSF CARGO DMS Georgia Tech, May 2004 When is a point of bS smooth, regular? How to check for r-regularity at B? –Check whether offset points is at distance r from bS Dist(B±rN,bS)

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APES: Ghrist, Rossignac, Szymczak, Turk11 NSF CARGO DMS Georgia Tech, May 2004 Mortar (definition) Core = R r (S) Mortar: M r (S ) = F r (S) – R r (S) Anticore = F r (S) Rounding: R r (S) Removes the red Fillet: F r (S) Adds the green Original set S The plane is divided into core, mortar, and aniticore

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APES: Ghrist, Rossignac, Szymczak, Turk12 NSF CARGO DMS Georgia Tech, May 2004 Example of Mortar Original shape Core Mortar

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APES: Ghrist, Rossignac, Szymczak, Turk13 NSF CARGO DMS Georgia Tech, May 2004 Properties of the mortar All points of the mortar are closer than r to the boundary bS –Restricting the effect of simplification to the mortar will ensure that we do not modify the shape in places far from its boundary The mortar excludes r-regular regions –Restricting the effect of simplification to the mortar will ensure that regular portions of the boundary are not affected by simplification SM r (S)M 2r (S)

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APES: Ghrist, Rossignac, Szymczak, Turk14 NSF CARGO DMS Georgia Tech, May 2004 The mortar is the fillet of the boundary Theorem: M r (S) is the topological interior of F r (bS) –Remove lower-dimensional (dangling) portions S bS M r (S) i(F r (bS))

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APES: Ghrist, Rossignac, Szymczak, Turk15 NSF CARGO DMS Georgia Tech, May 2004 Using the Mortar to decompose bS Given r, the regular segments of S are defined as the connected components of r-regular points of bS. Th1: Regular segment = connected component of bS–M r (S) Places where the core and anticore touch Th2: Irregular segment = connected component of bS M r (S) Note that irregular points may still be r-smooth Core = R r (S) M r (S) = F r (S) – R r (S) Regular segments

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APES: Ghrist, Rossignac, Szymczak, Turk16 NSF CARGO DMS Georgia Tech, May 2004 Mortar for multi-resolution analysis of space M r (S)M 2r (S) Regularity of a feature indicates its “thickness”

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APES: Ghrist, Rossignac, Szymczak, Turk17 NSF CARGO DMS Georgia Tech, May 2004 Analyzing the regularity of space The regularity of a point B with respect to a set S is defined as the minimum r for which B M r (S) –Points close to sharp features or constrictions are less regular –Different from signed distance field

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APES: Ghrist, Rossignac, Szymczak, Turk18 NSF CARGO DMS Georgia Tech, May 2004 Morphological Simplifications Fillet (closing) fills in creases and concave corners Rounding (opening) removes convex corners and branches Fillet and rounding operations may be combined to produce more symmetric filters that tend to smoothen both concave and convex features F r (R r (S)) and R r (F r (S)) combinations tend to: –Simplify topology: Eliminate small holes and components –Smoothen the shape almost everywhere –Regularize almost everywhere –Increase roundness (by reducing perimeter) However they –Do not guarantee r-regularity or r-smoothness –Tend to increase or to decrease the density Neither F r (R r (S)) nor R r (F r (S)) will make this set r-regular

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APES: Ghrist, Rossignac, Szymczak, Turk19 NSF CARGO DMS Georgia Tech, May 2004 Rounding and filleting combos F r (S)R r (F r (S))R r (S)SF r (R r (S)) F 2r (S) R 2r (F 2r (S))R 2r (S) F 2r (R 2r (S))

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APES: Ghrist, Rossignac, Szymczak, Turk20 NSF CARGO DMS Georgia Tech, May 2004 Which is better: FR or RF? F r (S) R r (F r (S)) R r (S) S F r (R r (S)) Removed by rounding Which option is better? The one that best preserves average density

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APES: Ghrist, Rossignac, Szymczak, Turk21 NSF CARGO DMS Georgia Tech, May 2004 The Mason filter We don’t have to make a global choice of FR versus RF Do it independently for each component of the mortar The Mason algorithm For each connected segment M of M r (S) replace M S by M F r (R r (S)) or by M R r (F r (S)), whichever best preserves the shape i.e., minimizes area of the symmetric difference between original and result “Mason: Morphological Simplification", J Williams, A Powell, and J Rossignac. GVU Tech. Report GIT-GVU Mason preserves density (average area) better than a global F r (R r (S)) or R r (F r (S)), but does not guarantee smoothness nor minimality of perimeter

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APES: Ghrist, Rossignac, Szymczak, Turk22 NSF CARGO DMS Georgia Tech, May 2004 Mason in Granada R r (F r (S)) S F r (R r (S))Mason Mortar M r (S): removed&added

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APES: Ghrist, Rossignac, Szymczak, Turk23 NSF CARGO DMS Georgia Tech, May 2004 Mason on a simple shape S Mason R r (F r (S)) F r (R r (S))

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APES: Ghrist, Rossignac, Szymczak, Turk24 NSF CARGO DMS Georgia Tech, May D Mason in China

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APES: Ghrist, Rossignac, Szymczak, Turk25 NSF CARGO DMS Georgia Tech, May 2004 Can we improve on Mason? Want to ensure r-smoothness Want to minimize perimeter Willing to give up some r-regularity Willing to give up some density preservation Formulate the solution using a Tight Hull… next slide

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APES: Ghrist, Rossignac, Szymczak, Turk26 NSF CARGO DMS Georgia Tech, May 2004 The tight hull, TH(R,F), of a set R inside a set F is the set H that has the smallest perimeter and satisfies R H F –Extends the idea of a convex hull, CH(R), which may be defined in 2D as the simply connected set that contains R and has smallest perimeter. Hence, CH(R)=TH(R,F) where F is the whole plane. bH is the shortest path around R in F Tight Hulls R F

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APES: Ghrist, Rossignac, Szymczak, Turk27 NSF CARGO DMS Georgia Tech, May 2004 Related prior art O(n) algorithm exists based on the Visibility graph – “Euclidean shortest path in the presence of rectilinear barriers”, D.T. Lee and. P. Preparata, Netwirks, 14: , 1984 –“Shortest paths and networks” Joe Mitchell, in Handbook of Discrete and Computational Geometry, Page 610, –“Shortest Paths in the Plane with Polygonal Obstacles" J Storer and J Reif Relative convex hull –Jack Sklansky and Dennis F. Kibler. A theory of nonuniformly digitized binary pictures. IEEE Transactions on Systems, Man, and Cybernetics, SMC-6(9): , –http://www.cs.mcgill.ca/~stever/pattern/MPP/node7.html#SECTION http://www.cs.mcgill.ca/~stever/pattern/MPP/node7.html#SECTION Minimal Perimeter Polygon –Steven M. Robbin, April 97 –http://www.cs.mcgill.ca/~stever/pattern/MPP/talk.htmlhttp://www.cs.mcgill.ca/~stever/pattern/MPP/talk.html

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APES: Ghrist, Rossignac, Szymczak, Turk28 NSF CARGO DMS Georgia Tech, May 2004 Properties of tight hulls Let H= TH(R,F) bH contains the portions of bR that are on the convex hull of R –bR bCH(R) bH bH is made of some convex portions of bR and of some concave portions of bF joined by short-cuts (straight edges) Edge (A,B) is a short-cut if A is a silhouette point for B and vice versa. R F

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APES: Ghrist, Rossignac, Szymczak, Turk29 NSF CARGO DMS Georgia Tech, May 2004 Start with CH(R), the convex hull of R Identify each edge (A,B) of CH(R) that is not on bR and that crosses bF Compute shortest path from A to B in F–R Computing tight hulls for polygons

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APES: Ghrist, Rossignac, Szymczak, Turk30 NSF CARGO DMS Georgia Tech, May 2004 Computing tight hulls for smooth shapes Shortest path –Reasonably easy for shapes bounded by lines and circular arcs. Track minimum distance field backwards –Propagate constrained distance field from A –Walk back from B along the gradient until you reach A Constrained curvature flow –Iterative smoothing (contraction) of boundary of F r (S) while preventing penetration in R r (S) and in the complement of F r (S) Morphological shaving (for discrete representations) –Grow core by adding to it straight line segments of contiguous mortar pixels that start and end at a core pixel

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APES: Ghrist, Rossignac, Szymczak, Turk31 NSF CARGO DMS Georgia Tech, May 2004 Tightening The tightening of a shape S is: T r (S) = TH(R r (S),F r (S)) –“Tightening: Perimeter-reducing, curvature-limiting morphological simplification", Jason Williams and Jarek Rossignac. In preparation. S

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APES: Ghrist, Rossignac, Szymczak, Turk32 NSF CARGO DMS Georgia Tech, May 2004 Example of tightening

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APES: Ghrist, Rossignac, Szymczak, Turk33 NSF CARGO DMS Georgia Tech, May 2004 Topological choices of tightening Invalid ?

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APES: Ghrist, Rossignac, Szymczak, Turk34 NSF CARGO DMS Georgia Tech, May 2004 Properties of tightenings bT r (S) is r-smooth T r (S) may have irregular parts

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APES: Ghrist, Rossignac, Szymczak, Turk35 NSF CARGO DMS Georgia Tech, May 2004 The road-tightening problem By law, the state must own all land located at a distance less or equal to r from a state road. The states owns an old road C and wants to make it r- smooth so it becomes a highway. Can it do so without purchasing any new land? –For simplicity, assume first that C is a manifold closed loop.

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APES: Ghrist, Rossignac, Szymczak, Turk36 NSF CARGO DMS Georgia Tech, May D tightening

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APES: Ghrist, Rossignac, Szymczak, Turk37 NSF CARGO DMS Georgia Tech, May 2004 The road-tightening solution Let S be the area enclosed by C. The new road will be bT r (S). –With some restrictions on C, we can extend this result to open road segments.

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APES: Ghrist, Rossignac, Szymczak, Turk38 NSF CARGO DMS Georgia Tech, May 2004 Comparisons simplified shapes R(F(S)) F(R(S)) mason tightening

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APES: Ghrist, Rossignac, Szymczak, Turk39 NSF CARGO DMS Georgia Tech, May 2004 FR, RF, mason, tightening Blue = Tightening Yellow = Core cyan = Mortar red = R(F(S)) Green = F(R(S)) brown = Mason

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APES: Ghrist, Rossignac, Szymczak, Turk40 NSF CARGO DMS Georgia Tech, May 2004 Summary and future work We propose to measure simplicity by regularity and smoothness –Defining regularity for all points of space will support a multi-resolution analysis of shape (interior, boundary, exterior) We restrict simplifications to the mortar, ensuring that regular areas are preserved –Mason improves on FR and RF combos by better preserving density –Tightening improves on Mason by minimizing perimeter and guaranteeing r-smoothness –We have applied it to the tightening of curves Future plans –Multi-resolution shape analysis and segmentation using regularity –Higher dimensions: surfaces, volumes, animations

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