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A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

A Nonlinear Contour Preserving 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

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

Problem description

Compression of images consisting of:

1

Smoothly coloured regions, separated by

2

Smooth contours In a 3D presentation the contours show up as discontinuities lying on a smooth curve in the domain.

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

Problem description

Compression of images consisting of:

1

Smoothly coloured regions, separated by

2

Smooth contours In a 3D presentation the contours show up as discontinuities lying on a smooth curve in the domain.

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

Horizon Class

Definition (Horizon Class H) H = {fΩ | fΩ(x, y) := 0c(x)>y 1c(x)≤y c(x) ∈ C2

Ω = = [0, 1]2 }

Remark Horizon Class Images are completely defined by c(x)

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

Why not use wavelets?

Approximation rate on Horizon Class Images

Tensor product wavelet approximants come from spaces Vn ⊂ Vn+1 on fixed partions. The number of rectangle subdomains cut by c(x) rize exponential w.r.t. j: nj ≈ O(2j). consequence: the total number of nonzero (significant) coefficients NJ = J

j=0 nj = O(2J)

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

Requirements of better methods (beyond wavelets)

1

can compactly represent line singularities →adaptive domain partitioning

2

possess good compression properties → sparse representation of smooth contours and smooth surfaces

3

fast decoding speed low inverse transformation complexity (O(N) )

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

Existing methods

Linear methods Curvelets Contourlets 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

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Concept: normal curves

Normal Offsets

Properties

Normal Offsets

Concept

Wavelet prediction step (No Update) Normal offset step Vertical

• ffset:

differ- ence between prediction and function value Normal

• ffset:

signed length between prediction and piercing point

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Concept: normal curves

Normal Offsets

Properties

Normal Offsets already succesfully applied for: Smooth manifolds

Remeshing of smooth 3D surfaces [Guskov et. al] irregular mesh

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Concept: normal curves

Normal Offsets

Properties

Normal Offsets already succesfully applied for: Smooth manifolds

Remeshing of smooth 3D surfaces [Guskov et. al] semi-regular mesh + 90% scalar coefficients

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Concept: normal curves

Normal Offsets

Properties

Normal Offsets already succesfully applied for: Smooth manifolds

Remeshing of smooth 3D surfaces [Guskov et. al] Approximation of smooth curves [Daubechies et. al]

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Concept: normal curves

Normal Offsets

Properties

Smooth and non-smooth

The nonlinear and highly adaptive character of a normal approximation scheme allows it to mimic both smooth as well as non smooth behaviour, without prior knowledge about the function f . smooth piecewise smooth

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Concept: normal curves

Normal Offsets

Properties

Properties

Geometrical information The normal offset coefficients information about:

1

the function value (z) (the what)

2

AND their location (the where) (x, y) 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

Concept: normal curves

Normal Offsets

Properties

Properties

Geometrical information The normal offset coefficients information about:

1

the function value (z) (the what)

2

AND their location (the where) (x, y) 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

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

How to extend to 2D-setting?

Define the normal direction...

...using a le Loop or Butterfly algorithm as for the 3D surface case ν∗

j+1 = 3 8

• ν1,i

j

+ ν2,i

j

• + 1

8

• ν3,i

j

+ ν4,i

j

• global mesh refinement operators

to avoid edge flipping − → non-hierarchical triangulations − → compression difficulties

• r exception handling needed to

stay in the same patch.

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

How to extend to 2D-setting?

Define the normal direction...

as lying in the vertical plane while ⊥ on the edge Local refinement operators (Edge refinement) − → nested triangulations i.e. Tj ⊂ Tj+1 − → suited for compression

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Topology

Regular vs. irregular meshes ⇔ Smooth vs. Nonsmooth

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.

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Methodology

method image dependent extra storage α

• subdivision

connectivity

• small

adaptive yes 2 bits/triangle greater

• α is the rate of approximation σLp(f ) = O(n−α)

subdivision connectivity adaptive tesselation

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Normal Offsets: Summary

Taylor made for efficient geometric image compression

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

• f edge on the XY -plane) → local refinement

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Approximation rate

Normal decomposition & n-terms approximation algorithm w.r.t. to the L1 distance norm. Approximation rate w.r.t. L1: ||f − fn||L1 = 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

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Lossless encoding

Conclusion Results

Lossless encoding

Cross section of an Horizon Class Image

Z Y X y x pj,k pj,k+1 p

* j+1,2k

Pj,k Pj,k+1 P

* j+1,2k

y’ x’ f(X,Y) f (x,y)

e

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Lossless encoding

Conclusion Results

Lossless encoding

Cross section of an Horizon Class Image

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

• f the edge

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Lossless encoding

Conclusion Results

Lossless encoding

Cross section of an Horizon Class Image

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

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Lossless encoding

Conclusion Results

Vertical indices

Discrete setting introduces vertical offsets Distribution of the vertical offsets is a mixture of a Laplace distribution and dirac distribution.

−300 −200 −100 100 200 300 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

mixture vertical offset

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Lossless encoding

Conclusion Results

Probability density function of normal indices

−0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 1 2 3 4 5 6 7

Mapping of indices → codewords (Huffman encoder) Can eventually be followed by an arithmetic encoder.

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Lossless encoding

Conclusion Results

Compression: conclusion

Where lies the compression? Normal offset transform vector to scalar (in fact Z2) Approximation rate w.r.t. H thanks to Edge Locating property Entropy encoding because of a priori PDF of projection

• f normal indices.[Lossless]

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Part V Some pictures...

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Images with smoothly colored areas

Original Image Normal compressed im- age 0.005bits/pixel

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Images with smoothly colored areas

Triangulation Normal compressed im- age 0.005bits/pixel

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Images with smoothly colored areas

JPEG2000 compressed image 0.005bits/pixel Normal compressed im- age 0.005bits/pixel

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Closer look

normal offsets

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

Closer look

JPEG2000 (wavelets)

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

More complex images

Figure: domain partitioning ‘Lena’

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

More complex images

Figure: Normal Mesh representation ‘Lena’

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

More complex images

Figure: Normal Mesh approximation ‘Lena’

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

The End

Thank you for your attention!

A Nonlinear Contour Preserving Transform for Geometrical Image Compression Ward Van Aerschot

The End

Thank you for your attention! Questions?