A top-down construction scheme for irregular pyramids Romain Goffe 1 Luc Brun 2 Guillaume Damiand 3 1 SIC-XLIM, Universit´ e de Poitiers, CNRS, UMR6172, Bˆ atiment SP2MI, F-86962, Futuroscope Chasseneuil, France 2 GREYC, ENSICAEN, CNRS, UMR6072, 6 Boulevard du Mar´ echal Juin, F-14050, Caen, France 3 LIRIS, Universit´ e Lyon, CNRS, UMR5205, Universit´ e Lyon 1, F-69622, Villeurbanne, France February 12, 2009
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Introduction 1 Recalls 2 A top-down model 3 Operations 4 5 Results Conclusion and perspectives 6
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Framework Application FoGrImMi project Very large medical images (30GB) Image processing and segmentation Requirements Image representation Segmentation and manipulation of regions Focus of attention over interesting areas Definition of a data structure Topological: process regions Hierarchical: multi-resolution images Top-down: limit memory requirements
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Timeline Model Drawbacks Quadtrees Segmentation ⇒ Regular pyramids problems Irregular Only bottom-up ⇒ pyramids constructions ⇒ Definition of a top-down and topological framework for irregular pyramids
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Introduction 1 Recalls 2 A top-down model 3 Operations 4 5 Results Conclusion and perspectives 6
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Combinatorial maps Initial image Disconnected Disconnected Map faces edges dart Region Notions Dart: ∼ half-edge β 1 permutation: turns around a face β 2 involution: gives the other orientation of the edge
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Topological maps Requirements Represent any partition Describe adjacency and inclusion relationships Efficient processing algorithms Combination of models Minimal combinatorial map (topology representation) Interpixel matrix (geometry information) Tree of regions (inclusion relationships)
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Topological maps Topology Geometry Inclusion relationships R ∞ β 1 R ∞ R ∞ R 1 R 1 β 1 R 2 R 2 β 2 β 2 R 1 β 1 R 3 R 3 β 1 R 2 R 3 β 2 β 1 β 1 Model features Complete (topology and geometry) Minimal (number of cells) Unique (same partition ⇔ same map)
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Pyramids Simple graph pyramids Stack of successively reduced graphs Difficult to update after operations Combinatorial pyramids Stack of successively contracted combinatorial maps Only bottom-up models Whole initial partition encoded Top-down pyramids Only encode upper levels Focus of attention: adjust segmentation from first discernable features
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Introduction 1 Recalls 2 A top-down model 3 Operations 4 5 Results Conclusion and perspectives 6
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Goals and definitions Goals Top-down topological model No explicit encoding Causal structure Easy update of the model after splitting Definitions Pyramid ∼ stack of linked topological maps A level k is deduced from k − 1, applying splitting operations
Introduction Recalls A top-down model Operations Results Conclusion and perspectives The hierarchical data structure Up/Down relations: k G k+1 G Between darts Between regions
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Global construction process R 0 R 0 1 2 Main steps Create first map G 0
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Global construction process R 0 R 0 1 2 Main steps Create first map G 0 G 1 is a copy linked to G 0 R 1 R 1 1 2
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Global construction process G 0 R 0 R 0 1 2 Main steps Create first map G 0 G 1 is a copy linked to G 0 Split G 1 G 1 R 1 1
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Global construction process R 0 R 0 1 2 Main steps Create first map G 0 G 1 is a copy linked to G 0 Split G 1 R 1 R 1 Merge G 1 3 1 R 1 2
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Duplicating a level for each dart in G m create a copy in G m + 1 ⇒ geometry link it with G m ⇒ up/down relations (darts) β 1 and β 2 sewing ⇒ topology for each region in G m create a copy in G m + 1 ⇒ adjacency relations link it with G m ⇒ up/down relations (regions) fill in region relations ⇒ tree of regions
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Refining a level Algorithm 1 : Refining Split region foreach region R ∈ G k do Keep region if splitting criterion(R) is true then Merge regions Split( R ); Keep edge Merge( G k , merging criterion); Simplify the map; S plitting criterion: selects one region for burst M erging criterion: operates on a couple of regions with the same parent
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Introduction 1 Recalls 2 A top-down model 3 Operations 4 5 Results Conclusion and perspectives 6
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Splitting operation Key points R 1 Initial region
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Splitting operation 6 7 8 Key points 5 Initial region Split edges 1 4 3 2
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Splitting operation 15 16 Key points 14 9 Initial region Split edges 13 10 Insert dangling edges 12 11
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Splitting operation R 4 R 5 Key points Initial region Split edges Insert dangling edges R 3 R 2 Sew darts
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Splitting Operation Burst method involvements create one region/pixel costly ⇒ But it is necessary to traverse all pixels to compute colorimetric information on new regions
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Merging Operation: general case R 1 Key points Initial regions R 2
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Merging Operation: general case R 1 Key points Initial regions Turn off geometry R 2
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Merging Operation: general case R 1 Key points Initial regions Turn off geometry Relabel darts R 1
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Merging Operation: general case R 1 Key points Initial regions Turn off geometry Relabel darts Remove darts
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Merging Operation: general case R 1 Key points Initial regions Turn off geometry Relabel darts Remove darts Result (after simplify)
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Merging Operation: constraint Constraint Only merge regions resulting from the splitting of a same parent Test Does the shared edge R 0 R 0 G 0 R 0 R 0 1 2 1 2 have a parent ? R 1 G 1 R 1 1 1
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Merging Operation: particular case Multiple adjacency Merging can be independent of criterion in multi-adjacency situations R 2 R 1 Steps Multi-adjacency between R 1 and R 2
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Merging Operation: particular case Multiple adjacency Merging can be independent of criterion in multi-adjacency situations 1 Steps Shared edges 2
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Merging Operation: particular case Multiple adjacency Merging can be independent of criterion in multi-adjacency situations Steps Merging criterion
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Merging Operation: particular case Multiple adjacency Merging can be independent of criterion in multi-adjacency situations R 1 Steps Remove edge 1
Introduction Recalls A top-down model Operations Results Conclusion and perspectives Merging Operation: particular case Multiple adjacency Merging can be independent of criterion in multi-adjacency situations R 1 Steps Remove edge 2
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