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A Nonlinear Contour Preserving Transform for Geometrical Image Compression A Nonlinear Contour Preserving Ward Van Aerschot Transform for Geometrical Image Compression Ward Van Aerschot K.U.Leuven www.cs.kuleuven.be/ ∼ ward

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot Introduction Wavelet methods Beyond wavelets Part I Problem description

Problem description A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot Introduction Wavelet methods Compression of images consisting of: Beyond wavelets Smoothly coloured regions, 1 separated by Smooth contours 2 In a 3D presentation the contours show up as discontinuities lying on a smooth curve in the domain.

Problem description A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot Introduction Wavelet methods Compression of images consisting of: Beyond wavelets Smoothly coloured regions, 1 separated by Smooth contours 2 In a 3D presentation the contours show up as discontinuities lying on a smooth curve in the domain.

Horizon Class A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot Introduction Wavelet methods Definition (Horizon Class H ) Beyond wavelets H = { f Ω | � 0 c ( x ) > y f Ω ( x , y ) := 1 c ( x ) ≤ y c ( x ) ∈ C 2 Ω = � = [0 , 1] 2 } Remark Horizon Class Images are completely defined by c ( x )

Why not use wavelets? A Nonlinear Contour Preserving Transform for Geometrical Image Approximation rate on Horizon Class Images Compression Ward Van Aerschot Tensor product wavelet approximants come from spaces Introduction V n ⊂ V n +1 on fixed partions. Wavelet methods The number of rectangle subdomains cut by c ( x ) Beyond wavelets rize exponential w.r.t. j : n j ≈ O (2 j ). consequence: the total number of nonzero ( significant ) coefficients N J = � J j =0 n j = O (2 J )

Requirements of better methods (beyond A Nonlinear Contour Preserving Transform for Geometrical Image wavelets) Compression Ward Van Aerschot Introduction Wavelet methods Beyond wavelets can compactly represent line singularities 1 → adaptive domain partitioning possess good compression properties 2 → sparse representation of smooth contours and smooth surfaces fast decoding speed 3 low inverse transformation complexity ( O ( N ) )

Existing methods A Nonlinear Contour Preserving Transform for Geometrical Image Compression Linear methods Ward Van Aerschot Curvelets Introduction Wavelet methods Contourlets Beyond wavelets etc. uses directional filterbanks to catch line singularities. [-] redundant decomposition. Non linear methods Wedgelets Binary Space partitioning algorithms etc. Normal Offsets → adaptive domain partitioning !

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot Concept: normal curves Normal Offsets Properties Part II Normal Approximation :1D

Normal Offsets A Nonlinear Contour Preserving Transform for Geometrical Image Concept Compression Ward Van Aerschot Concept: normal curves Normal Offsets Wavelet prediction step Normal offset step Properties (No Update) Vertical offset : differ- Normal offset : signed ence between prediction length between prediction and function value and piercing point

Normal Offsets already succesfully A Nonlinear Contour Preserving Transform for Geometrical Image applied for: Smooth manifolds Compression Ward Van Aerschot Concept: normal curves Normal Offsets Properties Remeshing of smooth 3D surfaces [Guskov et. al] irregular mesh

Normal Offsets already succesfully A Nonlinear Contour Preserving Transform for Geometrical Image applied for: Smooth manifolds Compression Ward Van Aerschot Concept: normal curves Normal Offsets Properties Remeshing of smooth 3D surfaces [Guskov et. al] semi-regular mesh + 90% scalar coefficients

Normal Offsets already succesfully A Nonlinear Contour Preserving Transform for Geometrical Image applied for: Smooth manifolds Compression Ward Van Aerschot Remeshing of smooth 3D surfaces [Guskov et. al] Concept: normal curves Normal Offsets Properties Approximation of smooth curves [Daubechies et. al]

Smooth and non-smooth A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot Concept: normal curves The nonlinear and highly adaptive character of a normal Normal Offsets Properties approximation scheme allows it to mimic both smooth as well as non smooth behaviour, without prior knowledge about the function f . smooth piecewise smooth

Properties A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot Concept: normal curves Normal Offsets Geometrical information Properties The normal offset coefficients information about: the function value ( z ) (the what) 1 AND their location (the where) ( x , y ) 2 Behaviour w.r.t. singularities Piercing points are attracted towards singularities. No major advantage in 1D = ⇒ becomes important in 2D

Properties A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot Concept: normal curves Normal Offsets Geometrical information Properties The normal offset coefficients information about: the function value ( z ) (the what) 1 AND their location (the where) ( x , y ) 2 Behaviour w.r.t. singularities Piercing points are attracted towards singularities. No major advantage in 1D = ⇒ becomes important in 2D

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot Part III Normal Approximation :2D

How to extend to 2D-setting? A Nonlinear Contour Preserving Transform for Geometrical Image Define the normal direction... Compression Ward Van Aerschot ...using a le Loop or Butterfly algorithm as for the 3D surface case � � � � ν 1 , i + ν 2 , i ν 3 , i + ν 4 , i j +1 = 3 + 1 ν ∗ 8 8 j j j j global mesh refinement operators to avoid edge flipping �− → non-hierarchical triangulations �− → compression difficulties or exception handling needed to stay in the same patch.

How to extend to 2D-setting? A Nonlinear Contour Preserving Transform for Geometrical Image Define the normal direction... Compression Ward Van Aerschot as lying in the vertical plane while ⊥ on the edge Local refinement operators (Edge refinement) �− → nested triangulations i.e. T j ⊂ T j +1 �− → suited for compression

Topology A Nonlinear Contour Preserving Transform for Geometrical Image Regular vs. irregular meshes ⇔ Smooth vs. Nonsmooth Compression Ward Van Aerschot In the smooth setting the approximating meshes are semi-regular (of subdivision connectivity starting from an irregular basemesh). In the nonsmooth setting topological information does matter ! We have to avoid that edges cross contours ⇒ irregular meshes .

Methodology A Nonlinear Contour Preserving Transform for Geometrical Image Compression ◦ method image extra storage α Ward Van Aerschot dependent subdivision - - small connectivity adaptive yes 2 bits/triangle greater ◦ α is the rate of approximation σ L p ( f ) = O ( n − α ) subdivision connectivity adaptive tesselation

Normal Offsets: Summary A Nonlinear Contour Preserving Transform for Geometrical Image Taylor made for efficient geometric image compression Compression Ward Van Aerschot Features piercing points are attracted towards the contours adaptive interconnection �→ triangle edges line up against the contour in a tangential manner. edge refinement method ( projection of normal vector on the projection of edge on the XY -plane ) �→ local refinement

Approximation rate A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot Normal decomposition & n-terms approximation algorithm w.r.t. to the L 1 distance norm. Approximation rate w.r.t. L 1 : || f − f n || L 1 = O ( n − 1 )

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot Lossless encoding Conclusion Results Part IV Encoding of normal coefficients

Lossless encoding A Nonlinear Contour Preserving Transform for Geometrical Image Cross section of an Horizon Class Image Compression Ward Van Aerschot Lossless encoding Conclusion y Z Results y’ P j,k+1 f(X,Y) p j,k+1 x’ * P Y * j+1,2k p j+1,2k p j,k x f (x,y) P j,k e X

Lossless encoding A Nonlinear Contour Preserving Transform for Geometrical Image Cross section of an Horizon Class Image Compression Ward Van Aerschot Lossless encoding Conclusion Results H = difference of the function values of the endpoints of the edge d = distance between edge point and discontinuity l = distance between the locations of the endpoints of the edge

Lossless encoding A Nonlinear Contour Preserving Transform for Geometrical Image Cross section of an Horizon Class Image Compression Ward Van Aerschot Lossless encoding Conclusion Results Definition (Normal index) A normal index is the signed number of pixels between the location of the midpoint and piercing point. sufficient to define normal offset