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
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