Transcript Slide 1
Extended Grassfire Transform on Medial Axes of 2D Shapes Tao Ju, Lu Liu Washington University in St. Louis Erin Chambers, David Letscher St. Louis University Medial axis • The set of interior points with two or more closest points on the boundary – A graph that captures the protrusions and topology of a 2D shape – First introduced by H. Blum in 1967 • A widely-used shape descriptor – Object recognition – Shape matching – Skeletal animation Grassfire transform • An erosion process that creates the medial axis – Imagine that the shape is filled with grass. A fire is ignited at the border and propagates inward at constant speed. – Medial axis is where the fire fronts meet. Medial axis significance • The medial axis is sensitive to perturbations on the boundary – Some measure is needed to identify significant subsets of medial axis Medial axis significance • A mathematically defined significance function that captures global shape property and resists boundary noise is lacking – Local measures • Does not capture global feature – Potential Residue (PR) [Ogniewicz 92], Medial Geodesic Function (MGF) [Dey • Discontinuous at junctions • Sensitive to boundary perturbations – Erosion Thickness (ET) [Shaked 98] • Lacking explicit formulation 06] Medial axis significance • A mathematically defined significance function that captures global shape property and resists boundary noise is lacking – Local measures • Does not capture global feature – Potential Residue (PR) [Ogniewicz 92], Medial Geodesic Function (MGF) [Dey 06] • Discontinuous at junctions • Sensitive to boundary perturbations – Erosion Thickness (ET) [Shaked 98] • Lacking explicit formulation Medial axis significance • A mathematically defined significance function that captures global shape property and resists boundary noise is lacking – Local measures • Does not capture global feature – Potential Residue (PR) [Ogniewicz 92], Medial Geodesic Function (MGF) [Dey 06] • Discontinuous at junctions • Sensitive to boundary perturbations – Erosion Thickness (ET) [Shaked 98] • Lacking explicit formulation Medial axis significance • A mathematically defined significance function that captures global shape property and resists boundary noise is lacking – Local measures • Does not capture global feature – Potential Residue (PR) [Ogniewicz 92], Medial Geodesic Function (MGF) [Dey 06] • Discontinuous at junctions • Sensitive to boundary perturbations – Erosion Thickness (ET) [Shaked 98] • Lacking explicit formulation Shape center • A center point is needed in various applications – Shape alignment – Motion tracking – Map annotation Shape center • Definition of an interior, unique, and stable center point does not exist so far – Centroid • not always interior – Geodesic center [Pollack 89] • may lie at the boundary – Geographical center • not unique Shape center • Definition of an interior, unique, and stable center point does not exist so far – Centroid • not always interior – Geodesic center [Pollack 89] • may lie at the boundary – Geographical center • not unique Centroid Shape center • Definition of an interior, unique, and stable center point does not exist so far – Centroid • not always interior – Geodesic center [Pollack 89] • may lie at the boundary – Geographical center • not unique Centroid Geodesic center Shape center • Definition of an interior, unique, and stable center point does not exist so far – Centroid • not always interior – Geodesic center [Pollack 89] • may lie at the boundary Centroid Geodesic center – Geographical center • not unique Geographic center Contributions • Unified definitions of a significance function and a center point on the 2D medial axis – The function: capturing global shape, continuous, and stable – The center point: interior, unique, and stable • A simple computing algorithm – Extends Blum’s grassfire transform • Applications Intuition • Measure the shape elongation around a medial axis point – By the length of the longest “tube” that fits inside the shape and is centered at that point Tubes • Union of largest inscribed circles centered along a segment of the medial axis – The segment is called the axis of the tube – The radius of the tube w.r.t. a point on the axis is its distance to the nearer end of the tube 𝑟𝑡 𝑥 = min(𝑑 𝑥, 𝑦1 + 𝑅 𝑦1 , 𝑑 𝑥, 𝑦2 + 𝑅 𝑦2 ) 𝑑: geodesic distance 𝑅: distance to boundary 𝑅(𝑦1 ) 𝑦1 𝑟𝑡 (𝑥) 𝑥 𝑦2 𝑅(𝑦2 ) Tubes • Union of largest inscribed circles centered along a segment of the medial axis – The segment is called the axis of the tube – The radius of the tube w.r.t. a point on the axis is its distance to the nearer end of the tube • Infinite on loop parts of axis (there are no “ends”) 𝑥 EDF • Extended Distance Function (EDF): radius of the longest tube 𝐸𝐷𝐹 𝑥 = 𝑠𝑢𝑝𝑡 𝑟𝑡 (𝑥) Simply connected shape EDF • Extended Distance Function (EDF): radius of the longest tube 𝐸𝐷𝐹 𝑥 = 𝑠𝑢𝑝𝑡 𝑟𝑡 (𝑥) 𝐸𝐷𝐹(𝑥) 𝑥 Simply connected shape EDF • Extended Distance Function (EDF): radius of the longest tube 𝐸𝐷𝐹 𝑥 = 𝑠𝑢𝑝𝑡 𝑟𝑡 (𝑥) 𝐸𝐷𝐹(𝑥) 𝑥 Simply connected shape EDF • Extended Distance Function (EDF): radius of the longest tube 𝐸𝐷𝐹 𝑥 = 𝑠𝑢𝑝𝑡 𝑟𝑡 (𝑥) Simply connected shape 𝑥 𝐸𝐷𝐹(𝑥) EDF • Extended Distance Function (EDF): radius of the longest tube 𝐸𝐷𝐹 𝑥 = 𝑠𝑢𝑝𝑡 𝑟𝑡 (𝑥) Shape with a hole EDF • Properties – No smaller than distance to boundary • Equal at the ends of the medial axis – Continuous everywhere • Along two branches at each junction – Constant gradient (1) away from maxima Distance to boundary EDF • Properties – No smaller than distance to boundary • Equal at the ends of the medial axis – Continuous everywhere • Along two branches at each junction – Constant gradient (1) away from maxima EDF – Loci of maxima preserves topology • Single point (for a simply connected shape) • System of loops (for shape with holes) Distance to boundary EDF • Properties – No smaller than distance to boundary • Equal at the ends of the medial axis – Continuous everywhere • Along two branches at each junction – Constant gradient (1) away from maxima EDF – Loci of maxima preserves topology • Single point (for a simply connected shape) • System of loops (for shape with holes) Distance to boundary EMA • Extended Medial Axis (EMA): loci of maxima of EDF – Intuitively, where the longest fitting tubes are centered EMA • Extended Medial Axis (EMA): loci of maxima of EDF – Intuitively, where the longest fitting tubes are centered • Properties – Interior – Unique point (For simply connected shapes) Extended grassfire transform • An erosion process that creates EDF and EMA – Fire is ignited at each end 𝑥 of medial axis at time 𝑅(𝑥), and propagates geodesically at constant speed. Fire front dies out when coming to a junction, and quenches as it meets another front. – EDF is the burning time – EMA consists of • Quench sites • Unburned parts Extended grassfire transform • An erosion process that creates EDF and EMA – Fire is ignited at each end 𝑥 of medial axis at time 𝑅(𝑥), and propagates geodesically at constant speed. Fire front dies out when coming to a junction, and quenches as it meets another front. – EDF is the burning time – EMA consists of • Quench sites • Unburned parts • A simple discrete algorithm Extended grassfire transform • Can be combined with Blum’s grassfire – Fire “continues” onto the medial axis at its ends Comparison with PR/MGF • EDF and EMA are more stable under boundary perturbation PR and its maxima Comparison with PR/MGF • EDF and EMA are more stable under boundary perturbation EDF and EMA Relation to ET • Erosion Thickness (ET) [Shaked 98] – The burning time of a fire that starts simultaneously at all ends and runs at non-uniform speed 1/(1 − 𝑅′(𝑥)) – No explicit definition exists EDF • New definition – 𝐸𝑇 𝑥 = 𝐸𝐷𝐹 𝑥 − 𝑅(𝑥) – Simpler to compute – More intuitive: length of the tube minus its thickness ET Application: Pruning Medial Axis • Observation – The difference between EDF and the distance-to-boundary gives a robust measure of shape elongation relative to its thickness EDF EDF and boundary distance Application: Pruning Medial Axis • Two significance measures: relative and absolute difference of EDF and boundary distance (R) – Absolute diff (ET): “scale” of elongation – Relative diff: “sharpness” of elongation 𝐸𝐷𝐹 𝑥 − 𝑅(𝑥) • Preserving medial axis parts that are high in both measures 1 − 𝑅(𝑥)/𝐸𝐷𝐹(𝑥) Application: Pruning Medial Axis • Preserving medial axis parts that score high in both measures Application: Pruning Medial Axis • Preserving medial axis parts that score high in both measures Application: Shape alignment • Stable shape centers for alignment Centroid Maxima of PR EMA Application: Shape alignment • Stable shape centers for alignment Centroid Maxima of PR EMA Application: Boundary Signature • Boundary Eccentricity (BE): “travel” distance to the EMA – Travel is restricted to be on the medial axis 𝑝 𝐵𝐸 𝑃 = 𝑑 𝑥, 𝐸𝑀𝐴 + 𝑅(𝑥) EMA 𝑥 Application: Boundary Signature • Boundary Eccentricity (BE): “travel” distance to the EMA – Highlights protrusions and is invariant under isometry Shape 1 Shape 2 Matching Summary • New definitions of significant function and medial point over the medial axis in 2D – EDF(x): length of the longest tube centered at x – EMA: the center of the longest tube • Extending Blum’s grassfire transform to compute them • Future work: 3D? – New global significance function on medial surfaces – New definition of center curve (or curve skeleton)