Bijective Composite Mean Value Mappings Kai Hormann Universit - PowerPoint PPT Presentation

Bijective Composite Mean Value Mappings Kai Hormann Università della Svizzera italiana, Lugano joint work with Michael S. Floater & Teseo Schneider

Introduction  special bivariate interpolation problem  find mapping f between two simple polygons  bijective  linear along edges 1/36 MAIA 2013 – Erice – 30 September 2013

Motivation  image warping original image mask warped image 2/36 MAIA 2013 – Erice – 30 September 2013

Barycentric coordinates  functions with  partition of unity  linear reproduction  Lagrange property  interpolation of data f i given at v i 3/36 MAIA 2013 – Erice – 30 September 2013

Barycentric coordinates  special case n = 3  general case  homogeneous weight functions with  barycentric coordinates 4/36 MAIA 2013 – Erice – 30 September 2013

Examples  Wachspress (WP) coordinates  mean value (MV) coordinates  discrete harmonic (DH) coordinates 5/36 MAIA 2013 – Erice – 30 September 2013

Barycentric mappings source polygon target polygon 6/36 MAIA 2013 – Erice – 30 September 2013

Wachspress mappings  based on WP coordinates [Wachspress 1975]  bijective for convex polygons [Floater & Kosinka 2010] 7/36 MAIA 2013 – Erice – 30 September 2013

Wachspress mappings  based on WP coordinates [Wachspress 1975]  bijective for convex polygons [Floater & Kosinka 2010]  not bijective for non-convex target  not well-defined for non-convex source 8/36 MAIA 2013 – Erice – 30 September 2013

Mean value mappings  based on MV coordinates [Floater 2003]  well-defined for non-convex source  not bijective 9/36 MAIA 2013 – Erice – 30 September 2013

Mean value mappings  based on MV coordinates [Floater 2003]  well-defined for non-convex source  not bijective, even for convex polygons 10/36 MAIA 2013 – Erice – 30 September 2013

Barycentric mappings WP MV   convex → convex   convex → non-convex   non-convex → convex   non-convex → non-convex  general barycentric mappings [Jacobson 2012]  not bijective for non-convex target polygon 11/36 MAIA 2013 – Erice – 30 September 2013

Composite barycentric mappings 12/36 MAIA 2013 – Erice – 30 September 2013

Sufficient condition  is bijective, if [Meisters & Olech 1963]  and without self-intersection  f bijective on the boundary  13/36 MAIA 2013 – Erice – 30 September 2013

Perturbation bounds  move one to  f bijective, if with  move all to  f bijective, if with 14/36 MAIA 2013 – Erice – 30 September 2013

Composite barycentric mappings  continuous vertex paths  intermediate polygons  barycentric mappings 15/36 MAIA 2013 – Erice – 30 September 2013

Composite barycentric mappings  partition of [0,1]  composite barycentric mapping  f τ bijective, if max displacement with max gradient 16/36 MAIA 2013 – Erice – 30 September 2013

Composite barycentric mappings 17/36 MAIA 2013 – Erice – 30 September 2013

Composite mean value mappings  use mean value coordinates to define mappings f k  well-defined, as long as without self-intersections  bounded for  if is convex [Rand et al. 2012]  if is non-convex ⇒ future work  constant M * is finite  f τ bijective for uniform steps  continuous vertex paths ϕ i  m sufficiently large 18/36 MAIA 2013 – Erice – 30 September 2013

Vertex paths  by linearly interpolating [Sederberg at al. 1993]  edges lengths  signed turning angles  barycentre  orientation of one edge 19/36 MAIA 2013 – Erice – 30 September 2013

Adaptive binary partition checkInterval ( 0 , 1 ) J min = computeJmin ( , ) if J min ≤ 0 then c = ( + ) /2 τ = τ ∪ c checkInterval ( , c ) checkInterval ( c , ) end end τ = { 0, 1 } 20/36 MAIA 2013 – Erice – 30 September 2013

Adaptive binary partition checkInterval ( 0 , 1 ) J min = computeJmin ( 0 , 1 ) if J min ≤ 0 then c = ( + ) /2 τ = τ ∪ c checkInterval ( , c ) checkInterval ( c , ) end end τ = { 0, 1 } 21/36 MAIA 2013 – Erice – 30 September 2013

Adaptive binary partition checkInterval ( 0 , 1 ) J min = computeJmin ( 0 , 1 ) if J min ≤ 0 then c = ( 0 + 1 ) /2 τ = τ ∪ c checkInterval ( 0 , c ) checkInterval ( c , 1 ) end end τ = { 0, 0.5, 1 } 22/36 MAIA 2013 – Erice – 30 September 2013

Adaptive binary partition checkInterval ( 0 , 0.5 ) J min = computeJmin ( 0 , 0.5 ) if J min ≤ 0 then c = ( + ) /2 τ = τ ∪ c checkInterval ( , c ) checkInterval ( c , ) end end τ = { 0, 0.5, 1 } 23/36 MAIA 2013 – Erice – 30 September 2013

Adaptive binary partition checkInterval ( 0 , 0.5 ) J min = computeJmin ( 0 , 0.5 ) if J min ≤ 0 then c = ( + ) /2 τ = τ ∪ c checkInterval ( , c ) checkInterval ( c , ) end end τ = { 0, 0.5, 1 } 24/36 MAIA 2013 – Erice – 30 September 2013

Adaptive binary partition checkInterval ( 0.5 , 1 ) J min = computeJmin ( 0.5 , 1 ) if J min ≤ 0 then c = ( + ) /2 τ = τ ∪ c checkInterval ( , c ) checkInterval ( c , ) end end τ = { 0, 0.5, 1 } 25/36 MAIA 2013 – Erice – 30 September 2013

Adaptive binary partition checkInterval ( 0.5 , 1 ) J min = computeJmin ( 0.5 , 1 ) if J min ≤ 0 then c = ( 0.5 + 1 ) /2 τ = τ ∪ c checkInterval ( 0.5 , c ) checkInterval ( c , 1 ) end end τ = { 0, 0.5, 0.75, 1 } 26/36 MAIA 2013 – Erice – 30 September 2013

Adaptive binary partition checkInterval ( 0.5 , 1 ) J min = computeJmin ( 0.5 , 1 ) if J min ≤ 0 then c = ( 0.5 + 1 ) /2 τ = τ ∪ c checkInterval ( 0.5 , c ) checkInterval ( c , 1 ) end end τ = { 0, 0.5, 0.75, ..., 1 } 27/36 MAIA 2013 – Erice – 30 September 2013

Composite barycentric mapping 28/36 MAIA 2013 – Erice – 30 September 2013

Image warping comparison mean value 29/36 MAIA 2013 – Erice – 30 September 2013

Image warping comparison composite mean value 30/36 MAIA 2013 – Erice – 30 September 2013

Infinite barycentric mappings 31/36 MAIA 2013 – Erice – 30 September 2013

Infinite barycentric mappings 32/36 MAIA 2013 – Erice – 30 September 2013

Infinite barycentric mappings 33/36 MAIA 2013 – Erice – 30 September 2013

Infinite barycentric mappings 1.E+01 10 1.E+00 0 error 10 -1 1.E-01 10 -2 1.E-02 100 1000 m 34/36 MAIA 2013 – Erice – 30 September 2013

Infinite barycentric mappings  infinite barycentric mapping: 35/36 MAIA 2013 – Erice – 30 September 2013

Conclusion  construction of bijective barycentric mappings  composition of intermediate mappings  theoretical bounds on the displacement  real-time composite mean value mappings  construction of the adaptive binary partition  real-time GPU implementation  infinite composite mappings  natural inverse  empiric result of convergence 36/36 MAIA 2013 – Erice – 30 September 2013