Feature Extraction 7-1 Ronald Peikert SciVis 2007 - Feature Extraction
What are features? Features are inherent properties of data, independent of coordinate frames etc. Dimension of a feature: • 0: point feature (often defined by n equations for n coordinates) • 1: line-like feature ( n -1 equations) • 2: surface-like feature • etc etc. • n: region-type feature (typically defined by a single inequality) 7-2 Ronald Peikert SciVis 2007 - Feature Extraction
Region-type features A feature is often indicated by high or low values of a derived field. Example: vortical regions in a flow field have been defined by = ∇× ω x v x ( ) ( ) • large magnitude of vorticity ω x v x ( ) ( ) ⋅ ω x ω x v x v x ( ) ( ) ( ) ( ) ⋅ • high absolute helicity high absolute helicity or normalized helicity or normalized helicity ω x v x ( ) ( ) ∇ ⋅∇ x p ( ) • positive pressure Laplacian ∇ v x ( ) • positive second invariant of the velocity gradient ( ( ) ) 2 ∇ ∇ + ∇ + ∇ v x v x 2 v x v x ( ) T ( ) ( ) ( ) • two negative eigenvalues of 2 The latter three definitions are parameter-free (preferred in feature p (p definitions). 7-3 Ronald Peikert SciVis 2007 - Feature Extraction
Point features in scalar fields Point features in scalar fields: • local minima/maxima • saddle points saddle points ∇ = x s ( ) 0 occur at zero gradient ( n scalar equations), (places where height field is horizontal). The above point features are the places where the contour line or isosurface changes its topology when the level is varied from min to max. i t The contour tree (or Reeb ( graph) describes the split and join events. Image credit: S. Dillard 7-4 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields Line-like features in 2D scalar field: Watersheds describe ridges/valleys of a height field s( x ): ∇ x integrate the gradient field s ( ) (backward/forward), starting at saddle points. The watersheds provide a segmentation of the domain into so- called Morse-Smale complexes. streamlines of streamlines of (negative) pass gradient field (saddle) peak peak pit pit (local max) (local min) pass (saddle) 7-5 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields Watersheds require integration, are therefore not locally detectable. Alternative definition of ridges/valleys (in n D scalar fields)? Local minima/maxima: Known at least since differentiation was invented (17 th century)! • • What is the natural extension to 1D? 7-6 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields Often used concepts: • profile-based ridges • curvature extrema on height contours Counter-example for both (Wiener 1887!): "inclined elliptic cylinder" 7-7 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields Question: How are local maxima most naturally extended to 1D features? Answer: height ridges. Surprisingly, a formal definition of height ridges was given only in the 1990s (Eberly, Lindeberg), based on Haralick's definition (1983). In contrast, local minima/maxima are known for centuries. De Saint-Venant (1852) defined a concept similar to height ridges. Image credit: P. Majer 7-8 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields At a given point x 0 the scalar field has the Taylor approximation ( ( ) ) ( ( ) ) ( ( ) ) 3 + = + ∇ ⋅ + + x x x x x Hx T x s s s O 0 0 0 0 where H is the Hessian matrix of second derivatives ⎛ ⎛ ⎞ ⎞ 2 ( ) ∂ x 2 s = ⎜ H ⎟ ⎜ ⎟ ∂ ∂ x x ⎝ ⎠ i j ij H has real eigenvalues and orthogonal eigenvectors. By taking the eigenvectors as the coordinate frame, H becomes the diagonal matrix diagonal matrix λ ⎛ ⎞ 0 1 ⎜ ⎟ = ⎜ H � ⎟ ⎜ ⎜ ⎟ ⎟ λ ⎝ 0 ⎠ n 7-9 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields R ∈ A point is a local maximum of s( x ) if for all n axes: x n = = = � s s 0 • the first derivatives are zero: the first derivatives are zero: x x x x 1 1 n < � s 1 , , s 0 • the second derivatives are negative: x x x x 1 n n In the appropriate coordinate frame, this generalizes to: R ∈ A point is on a d -dimensional height ridge of s( x ) if for the x n first n - d axes: = = = � s s 0 • first derivatives are zero: x x − 1 n d < � • second derivatives are negative: s 1 , , s 0 x x x x x x x x − − 1 1 1 n d n d n d n d 7-10 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields Appropriate coordinate frame means: axes are aligned with eigenvectors of H • λ ≥ ≥ λ � • ordered by absolute eigenvalues: 1 n Remark: We used Lindeberg's definition. In Eberly's definition λ ≤ ≤ λ � axes are ordered by signed eigenvalues: 1 n This is slightly weaker (accepting more points). Example: scalar field, (1D) height ridge according to Eberly and E l l fi ld (1D) h i ht id di t Eb l d Lindeberg: image credit: P. Majer 7-11 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields λ λ ≥ ≥ λ λ Sk t h Sketch of cases (with Lindeberg's definition, ) f ( ith Li d b ' d fi iti ) 1 2 none points on points on valley lines ll li ridge lines λ 2 < 0 λ 2 > 0 λ 2 ∇ s( x ) ∇ s( x ) ∇ ( ) ∇ s( x ) ∇ s( x ) λ 1 < 0 λ 1 > 0 λ 1 λ 2 < 0 λ 2 λ 2 > 0 ∇ s( x ) ∇ s( x ) ∇ s( x ) λ 1 > 0 λ 1 < 0 λ 1 7-12 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields "Circular gutter" example (Koenderink / van Doorn): Height field in polar coordinates: Height field in polar coordinates: ( ) ( ) 2 ϕ = − + ϕ z r , 1 r k • k describes the steepness in the k describes the steepness in the tangential direction. • Profiles in radial sections are parabolas: ( ) ( ) 2 ϕ = − + z r , 1 r const • Lowest points in sections ϕ = const lie on the r = r = 1 1 Circular gutter (with two Circular gutter (with two asymptote circle asymptote circle . straight segments added) 7-13 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields Circular gutter: Height ridge deviates Height ridge deviates (in the circular part) height ridge from the solution (of negative field) ci ircular va given by radial watershed profiles. (of negative field) alley h i ht height contours t "Counter-example" slope lines for height ridges. 7-14 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields Blended height fields (replacing the circular part by a blend of the part by a blend of the two height fields): height ridge (of negative field) b lended fie Watershed deviates watershed (in the lower part) (of negative field) from obvious from obvious elds height contours h i ht t symmetric valley line. slope lines "Counter-example" for watersheds. 7-15 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields The use of watersheds vs. height ridges is still heavily discussed in computer vision (Koenderink/van Doorn '93). Watersheds: Watersheds: + are slope lines of height field (=streamlines of gradient field) - depend on boundaries - require existence of a saddle point point watershed no watersheds Image credit: C. Steger 7-16 Ronald Peikert SciVis 2007 - Feature Extraction
Line-like features in scalar fields Height ridges in 3D scalar fields can be used for defining/detecting vortex core lines. These are • by Kida and Miura: height ridges (valley lines) of pressure by Kida and Miura: height ridges (valley lines) of pressure • • by Ahmad/Kenwright/Strawn: height ridges of vorticity magnitude by Ahmad/Kenwright/Strawn: height ridges of vorticity magnitude 7-17 Ronald Peikert SciVis 2007 - Feature Extraction
Geometric features of surfaces On surfaces in 3-space, 0- and 1-dimensional features can be defined by the (differential) geometry alone. Geometric features vs features of a field Geometric features vs. features of a field. Examples of geometric features (not a core subject of SciVis), κ κ κ ≥ κ based on principal curvatures , , 1 2 1 2 κ = κ • umbilic points: 1 2 κ κ • curvature ridges: Loci of points where curvature ridges: Loci of points where is a maximum along is a maximum along 1 the associated curvature line Image credit: Y. Ohtake 7-18 Ronald Peikert SciVis 2007 - Feature Extraction
Geometric features of surfaces The term "ridge" can refer to either height ridges or curvature ridges. Curvature ridges are not appropriate as features of a scalar field Curvature ridges are not appropriate as features of a scalar field (height field). Reason: Invariance under rotation (tilting). 7-19 Ronald Peikert SciVis 2007 - Feature Extraction
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