Diego Nehab. n A Transformation For Extracting New Descriptors of Shape. n Locus of points equidistant from contour

Transcription

1 Diego Nehab A Trasformatio For Extractig New Descriptors of Shape Locus of poits equidistat from cotour Medial Axis Symmetric Axis Skeleto Shock Graph Shaked 96 1

2 Shape matchig Aimatio Dimesio reductio Solid modelig Smoothig or sharpeig of shape Motio plaig Mesh geeratio Defiitios, properties, ad examples Applicatio examples How to compute Hierarchic Vorooi Skeletos No coclusios 2

3 Locus of poits equidistat from cotour Grass-fire, prairie-fire, wave-frot collisio Locus of ceters of maximal circles Local maxima i distace trasform Result of topology preservig thiig Ridges i evelope of coes (apexes o cotour) Grass-fire, prairie-fire, wave-frot collisio Locus of ceters of maximal circles va Toder 3

4 Medial Axis augmeted by radius fuctio Trasformatio is ivertible Equidistat from 1 joit pit set (ed poits) Equidistat from 2 disjoit sets (ormal poits) Equidistat from 3 or more disjoit sets (brach poits) Sebastia 01 4

5 Aalyze skeleto evolutio i time, flow of shocks Split skeleto ito mootoic segmets More refied tha MA Blum 67 Defiitios, properties, ad examples Applicatio examples How to compute Hierarchic Vorooi Skeletos No coclusios 5

6 !"# Staff-lie detectio of music scores Compute medial axis of score Extract ad simplify polygoal graph Compute histogram of segmet directios Project segmets i the most popular directio Detect peaks i resultig histogram!"# 6

7 !"$ Recogitio of Shapes by Edittig Shock Graphs Compute Shock Graphs for each shape Shapes whose shock graphs have same topology are clustered ito equivalet classes Editig operatios are trasitios betwee classes Associate a cost to each edit operatio Fid miimum edit cost path betwee shapes!"$ 7

8 Defiitios, properties, ad examples Applicatio examples How to compute Hierarchic Vorooi Skeletos No coclusios %& Liear Skeletos from Square Cupboards Work iwards from the boudary Remove all poits except for skeleto poits Preserve topology by a umber of tests '((( ) Repeat util o poit ca be removed 8

9 %* Distace i shape with respect to it's complemet Copy of shape, each poit is labeled with distace Each poit represets disk cetered at the poit Disk size is give by poit label Disk shape is give by distace metric Ca be computed i two passes over image 9

10 Chamfer distaces are easier to use tha Euclidea +, (, -.!+$($. /!+$(. But ustable with rotatio (3,4)-DT is good compromise

11 1&2 Well-Shaped, Stable ad Reversible Skeletos from the (3,4)-Distace Trasform Compute (3,4) DT Idetify local maxima ad saddle poits Grow coectig paths i directio of maximal gradiet Hole fill, fial thi 3 Disk is maximal if ot cotaied by ay other Ceter of MD is a local maximum of DT Label compariso betwee eighbors is eough to determie cotaimet Disk is maximal if it is ot the smaller eighbor of ay of its eighbors 11

12 4/&5 Simulatig the Grassfire Trasform usig a Active Cotour Model Compute the Distace Trasform Defie a potetial fuctio equal to -DT Place a sake over the iso-cotour -1 Iterate based o gradiet ad iteral costraits Skeleto poits correspod to Sake meetig poits 4/&5 12

13 & The Lie Skeleto Defiitios, properties, ad examples Applicatio examples How to compute Hierarchic Vorooi Skeletos No coclusios 13

14 6 7* Skeletos are highly sesitive to oise i boudary Regularizatio ca be performed i shape (smoothig) or i MAT (pruig) Pruig affects the iverse MAT ad smoothig affects the MAT 14

15 Discrete case, apply hole-fillig before, Curvature flow Blur image Usually, pruig skeleto is better idea &(8 9 / Small perturbatios i the axis or the associated fuctio may result i a axis-like descriptio that does ot correspod to a plaar shape Simple trasformatios kow to be allowed are :-!(--/ 15

16 ;*&< Hierarchic Vorooi Skeletos Compute VD of shape s boudary poits Associate topology preservig importace metric with each edge of VD Hierarchic clusterig of skeleto edges creates a skeleto pyramid Geodesic distace betwee geerators of a give edge Assig to each edge their correspodig achor distace For ay edge, there is a mootoic path util max Thresholdig matais coectivity Ogiewicz 95 16

17 6 Ogiewicz 95 %6 Clea with achor distace threshold Start with first level edges ; Follow skeleto i order of least steep descet ; ; - 17

18 Defiitios, properties, ad examples Applicatio examples How to compute Hierarchic Vorooi Skeletos No coclusios Blum 67 Hilditch 69 Bookstei 79 Leymarie 92 di Baja 94 Ogiewicz 95 Sebastia 01 18