Performance Evaluation for Scattered Data Interpolation
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Performance Evaluation for Scattered Data Interpolation
Matthew P. Foster & Adrian N. [email protected]
University of Bath
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Summary
Basics Performance Outputs
• Scattered data
• Interpolation
• Basics
• Methods
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Summary
Basics Performance Outputs
• Performance Evaluation
• Simulation-validation
• Cross-validation
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Summary
Basics Performance Outputs
• Output Evaluation
• Error distributions
• Differences & artefacts
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Scattered Data & Interpolation
• 2-D (+ height) in this case:
• x, y, z triplets
• Or matrix projections
• Very common
• Common examples:
• Nearest Neighbour
• Linear, Cubic
• Kriging
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Interpolation Methods
• All techniques fit into two classes:
• Local – points in neighbourhood
• Global – all points
• Both use weighted combinations of input points, the weightings can be based on:
• Geometry – ‘where?’
• Input characteristics – ‘what?’
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Point Geometry
• Delaunay Triangulation / Voronoi diagram
• Arguably most fundamental
• Distance metric
• Or scale-space version
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Image Characteristics
• Correlation
• E.g. Semivariogram
• Local image information
• Energy
• Orientation
• Anisotropy
• Other methods…
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Image Characteristics
• Correlation
• E.g. Semivariogram
• Local image information
• Energy
• Orientation
• Anisotropy
• Other methods…
Local orientation
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Method Locality Weighting Method
ANC LocalApplicability
function Steered filters
Kriging Global Basis functionBuild model
then fit
Linear / Cubic Local Triangulation Surface fitting
Natural Neighbour Local Triangulation
Area Weighting
RBF Global Basis function Linear fitting
Methods
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Performance Evaluation
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Simulation-Validation
• Workflow
• Generate
• Sample
• Interpolate
• Subtract
• Repeat
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0
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Sparsity
ANCCubicKrigingNat. Neighbour
Simulation-Validation
• Give a good ‘feel’ for performance
• Detailed analysis possible
• Rarely mirrors actual data
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Cross-Validation
• Computer vision / classification technique
• Allow performance analysis using real data
• Partition into 2 classes
• Reconstruction
• Validation
180
o W
135oW
90
oW
45 o
W
0 o
10 o
S
0 o
10 oN
20 oN
30 oN
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o N
60
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oN
80 o
N
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Example: TEC Data
• Data from GPS Satellites
• During Halloween Storm -Oct. 2003
• Fairly sparse relative to field size 100 x 120 (0.5˚)
180o W
135 oW
90 oW
45 oW
0 o
10 oS
0 o
10 oN
20 oN
30 oN
50
o N 6
0oN
70o N 80 oN
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Process• For each time interval
• Split in 10 random blocks
• Reconstruct using 1-9 blocks
• Validate with remaining blocks
• Repeat as necessary Cubic
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Results
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Prop
ortio
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MSE
Sparsity
ANCCubicKrigingNatural Neighbour
• Noisier than simulations
• Some similarities
• Kriging peak
• General method performance
See: An Evaluation of Interpolation Methods for Ionospheric TEC Mapping, M. P. Foster and A. N. Evans. IEEE Trans. Geoscience and Remote Sensing. Vol 46, No. 7, pp. 2153 -
2164, 2008
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Output Evaluation
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Error Value
Error Histogram
Error Distributions
• When everything works, outputs look nice
• Histogram is approximately Gaussian
Kriging Reconstruction
Fractal Surface
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Error Histogram
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Sem
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lag [h]
Semivariogram with Fitted Spherical Model
Error Distributions
• When everything works, outputs look nice
• Histogram is approximately Gaussian
Kriging Reconstruction
Fractal Surface
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Error Value
Error Histogram
Error Distributions
• When it doesn’t work well:
• Examining histogram can show problems
• Which you can then look into:
• bad model fitting (due to odd image!)
Kriging Reconstruction
Image Data
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Error Histogram
0
0.05
0.1
0.15
0.2
0 10 20 30 40 50 60 70
Sem
ivaria
nce
[γ (h
)]
lag [h]
Semivariogram with Fitted Spherical Model
Error Distributions
• When it doesn’t work well:
• Examining histogram can show problems
• Which you can then look into:
• bad model fitting (due to odd image!)
Kriging Reconstruction
Image Data
![Page 23: Performance Evaluation for Scattered Data Interpolation](https://reader034.fdocuments.us/reader034/viewer/2022052304/55796bcad8b42a3a5c8b4cf5/html5/thumbnails/23.jpg)
Artefacts
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LinearShuttle Radar Topography Mission Reconstructed
from ~1% of samples
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Linear RBFReconstructed from ~1% of
samples
Shuttle Radar Topography Mission
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TPS RBFReconstructed from ~1% of
samples
Shuttle Radar Topography Mission
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Natural Neighbour
Reconstructed from ~1% of
samples
Shuttle Radar Topography Mission
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ANCReconstructed from ~1% of
samples
Shuttle Radar Topography Mission
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Artefacts
NaturalNeighbour
TPS (Cubic)
Cubic
Linear
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Artefacts
NaturalNeighbour
TPS (Cubic)
Cubic
Linear
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Artefacts
PointyNatural
Neighbour
TPS (Cubic)
Cubic
Linear
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Artefacts
PointyNatural
Neighbour
TPS (Cubic)
Cubic
Linear
Overshoot
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Artefacts
PointyNatural
Neighbour
TPS (Cubic)
Cubic
Linear
Overshoot
Triangulation Edges
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Conclusions• Quantitative methodologies are useful for analysing
performance
• Result from real data can be very different from simulations
• But don’t yield information about spatial error distribution, or artefacts produced by different methods
• Error distributions can be used for more detailed qualitative analysis, provided enough data are available.
• The method best method depends on the application.