Basis function Wavelet function · Haar wavelets Basis function Wavelet function. 320491: Advanced...
Transcript of Basis function Wavelet function · Haar wavelets Basis function Wavelet function. 320491: Advanced...
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Haar wavelets
Basis function
Wavelet function
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Haar wavelets
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Multiresolution representation
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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.
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Multiresolution representation with Haar wavelets
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Memory requirements
Storing the highest resolutionrequires the same amount of storageas storing the coarsest resolutionand all detail coefficients
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1.4.2 Wavelets
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References
• Wavelets for Computer Graphics: A Primer.Eric J. Stollnitz, Tony D. DeRose, David H. SalesinUniversity of Washington, technical report, September 1994.
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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:
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Multiresolution discrete wavelet transform• Basis and wavelet functions span spaces:
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Multiresolution discrete wavelet transform
• The spaces form a multiresolution analysis:
• The spaces are the orthogonal complements:
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Constant B-spline wavelets (Haar wavelets)
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Constant B-spline wavelets (Haar wavelets)
The wavelet transform can also be written in matrix form:
The matrices are called synthesis matrices.
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Linear B-spline wavelets
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Linear B-spline wavelets
Synthesis matrices:
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Quadratic B-spline wavelets
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Quadratic B-spline wavelets
Synthesis matrices:
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Quadratic B-spline wavelets
Synthesis matrices:
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Cubic B-spline wavelets
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Cubic B-spline wavelets
Synthesis matrices:
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Cubic B-spline wavelets
Synthesis matrices:
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2D Haar wavelet transform
• 2D basis and wavelet functions are tensor productsof 1D basis and wavelet functions.
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2D Haar wavelet transformBasis:
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2D Haar wavelet transform
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2D Haar wavelet transformAlternative construction:
Use 2D basis function
and three 2D wavelet functions
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2D Haar wavelet transformBasis:
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2D Haar wavelet transform
Advantage: One obtains undistorteddownscaled versions of the 2D image.
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2D wavelet transform in RGB space
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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.
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Image compression
100% 21% 4% 1%Error: 0% 5% 10% 15%
Haar wavelets:
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Image compressionJPEG 2000: Cohen-Daubechies-Feauveau wavelets
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Image compressionJPEG 2000: lossy compression leads to blurring.
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1.4.3 Multiresolution Modeling
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References
• Multiresolution Techniques. Leif P. Kobbelt. In: Handbook of Computer Aided Geometric Design, G. Farin, J. Hoschek, M-S. Kim (eds.), Elsevier, 2002.
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Multiresolution modeling
• Filter bank
small details large detailslow frequencyhigh frequency
frequency spectrum
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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.
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Multiresolution modeling: Example 1
• Global vs. local details:
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Multiresolution modeling: Example 2
• Change the character of an object by replacing detail coefficients with new ones, e.g., by replacing smallhigh-frequency details with large ones.
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Multiresolution modeling: Example 3
• Performing interactive changes of the shape at different levels of resolution.
fine level intermediate level coarse level
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Multiresolution modeling: Example 3• The finer the level of resolution, the more local the
change.
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• • •
ApplicationsHierarchy:
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~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:
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Level of detail controlDisplay at appropriatelevel of detail:
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Progressive transmission and renderingProgressive refinement of mesh over time:
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Multiresolution editingShape editing at different LODs: