basis function wavelet function · haar wavelets basis function wavelet function. 320491: advanced...

320491: Advanced Graphics - Chapter 1 146

Visualization and Computer Graphics LabJacobs University

Haar wavelets

Basis function

Wavelet function

Haar wavelets

Multiresolution representation

• Object is represented as a sequence of resolutions.• The resolutions are called levels (levels of detail,

LOD)• The differences are called detail coefficients.• The levels build a multiresolution hierarchy:

• The level is the base level.• The base level does not need to be represented by a

regular mesh. All levels use then semi-regular meshes.

Multiresolution representation with Haar wavelets

Memory requirements

Storing the highest resolutionrequires the same amount of storageas storing the coarsest resolutionand all detail coefficients

1.4.2 Wavelets

References

• Wavelets for Computer Graphics: A Primer.Eric J. Stollnitz, Tony D. DeRose, David H. SalesinUniversity of Washington, technical report, September 1994.

Wavelets• Haar wavelets are the simplest and oldest wavelets

(Alfred Haar, 1909).• Wavelet theory started much later (boom in the 80s).• Wavelet functions must fulfill the criteria:

• Examples:

Multiresolution discrete wavelet transform• Basis and wavelet functions span spaces:

Multiresolution discrete wavelet transform

• The spaces form a multiresolution analysis:

• The spaces are the orthogonal complements:

Constant B-spline wavelets (Haar wavelets)

The wavelet transform can also be written in matrix form:

The matrices are called synthesis matrices.

Linear B-spline wavelets

Synthesis matrices:

Quadratic B-spline wavelets

Synthesis matrices:

Cubic B-spline wavelets

Synthesis matrices:

2D Haar wavelet transform

• 2D basis and wavelet functions are tensor productsof 1D basis and wavelet functions.

2D Haar wavelet transformBasis:

2D Haar wavelet transformAlternative construction:

Use 2D basis function

and three 2D wavelet functions

2D Haar wavelet transformBasis:

Advantage: One obtains undistorteddownscaled versions of the 2D image.

2D wavelet transform in RGB space

Image compression

• Loss-less compression– Do not store detail coefficients that are 0.– Constant regions are stored by 1 value only.

• Lossy compression– Set detail coefficients with small absolute values to 0.– A threshold determines the compression rate.

Image compression

100% 21% 4% 1%Error: 0% 5% 10% 15%

Haar wavelets:

Image compressionJPEG 2000: Cohen-Daubechies-Feauveau wavelets

Image compressionJPEG 2000: lossy compression leads to blurring.

1.4.3 Multiresolution Modeling

References

• Multiresolution Techniques. Leif P. Kobbelt. In: Handbook of Computer Aided Geometric Design, G. Farin, J. Hoschek, M-S. Kim (eds.), Elsevier, 2002.

Multiresolution modeling

• Filter bank

small details large detailslow frequencyhigh frequency

frequency spectrum

Multiresolution modeling: Example 1

1. Remove high-frequency detail by going to coarseresolution.

2. Perform modeling at coarse resolution byinteractively changing the shape.

3. Reinsert high-frequency detail by going to fine resolution.

• Global vs. local details:

• Change the character of an object by replacing detail coefficients with new ones, e.g., by replacing smallhigh-frequency details with large ones.

• Performing interactive changes of the shape at different levels of resolution.

fine level intermediate level coarse level

Multiresolution modeling: Example 3• The finer the level of resolution, the more local the

change.

• • •

ApplicationsHierarchy:

~70,000 faces ~11,000 faces~70,000 faces ~11,000 faces

ε < 0.8%ε < 0.8%

CompressionReduction of storage requirements with error bounds:

Level of detail controlDisplay at appropriatelevel of detail:

Progressive transmission and renderingProgressive refinement of mesh over time:

Multiresolution editingShape editing at different LODs:

basis function wavelet function · haar wavelets basis function wavelet function. 320491: advanced...

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